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Theorem fin1a2lem13 9426
 Description: Lemma for fin1a2 9429. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fin1a2lem13 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)

Proof of Theorem fin1a2lem13
Dummy variables 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 479 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → (𝐵𝐶) ∈ FinII)
2 simpll1 1255 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐴 ⊆ 𝒫 𝐵)
3 ssel2 3739 . . . . . . . . . 10 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔 ∈ 𝒫 𝐵)
43elpwid 4314 . . . . . . . . 9 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → 𝑔𝐵)
54ssdifd 3889 . . . . . . . 8 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑔𝐶) ⊆ (𝐵𝐶))
6 sseq1 3767 . . . . . . . 8 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝐵𝐶) ↔ (𝑔𝐶) ⊆ (𝐵𝐶)))
75, 6syl5ibrcom 237 . . . . . . 7 ((𝐴 ⊆ 𝒫 𝐵𝑔𝐴) → (𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
87rexlimdva 3169 . . . . . 6 (𝐴 ⊆ 𝒫 𝐵 → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → 𝑓 ⊆ (𝐵𝐶)))
9 vex 3343 . . . . . . 7 𝑓 ∈ V
10 eqid 2760 . . . . . . . 8 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑔𝐴 ↦ (𝑔𝐶))
1110elrnmpt 5527 . . . . . . 7 (𝑓 ∈ V → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶)))
129, 11ax-mp 5 . . . . . 6 (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑓 = (𝑔𝐶))
13 selpw 4309 . . . . . 6 (𝑓 ∈ 𝒫 (𝐵𝐶) ↔ 𝑓 ⊆ (𝐵𝐶))
148, 12, 133imtr4g 285 . . . . 5 (𝐴 ⊆ 𝒫 𝐵 → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝑓 ∈ 𝒫 (𝐵𝐶)))
1514ssrdv 3750 . . . 4 (𝐴 ⊆ 𝒫 𝐵 → ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶))
162, 15syl 17 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶))
17 simplrr 820 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → 𝐶𝐴)
18 difid 4091 . . . . . . 7 (𝐶𝐶) = ∅
1918eqcomi 2769 . . . . . 6 ∅ = (𝐶𝐶)
20 difeq1 3864 . . . . . . . 8 (𝑔 = 𝐶 → (𝑔𝐶) = (𝐶𝐶))
2120eqeq2d 2770 . . . . . . 7 (𝑔 = 𝐶 → (∅ = (𝑔𝐶) ↔ ∅ = (𝐶𝐶)))
2221rspcev 3449 . . . . . 6 ((𝐶𝐴 ∧ ∅ = (𝐶𝐶)) → ∃𝑔𝐴 ∅ = (𝑔𝐶))
2319, 22mpan2 709 . . . . 5 (𝐶𝐴 → ∃𝑔𝐴 ∅ = (𝑔𝐶))
24 0ex 4942 . . . . . 6 ∅ ∈ V
2510elrnmpt 5527 . . . . . 6 (∅ ∈ V → (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶)))
2624, 25ax-mp 5 . . . . 5 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 ∅ = (𝑔𝐶))
2723, 26sylibr 224 . . . 4 (𝐶𝐴 → ∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
28 ne0i 4064 . . . 4 (∅ ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
2917, 27, 283syl 18 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅)
30 simpll2 1257 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or 𝐴)
31 vex 3343 . . . . . . . 8 𝑥 ∈ V
3210elrnmpt 5527 . . . . . . . 8 (𝑥 ∈ V → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶)))
3331, 32ax-mp 5 . . . . . . 7 (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 𝑥 = (𝑔𝐶))
34 difeq1 3864 . . . . . . . . . 10 (𝑔 = 𝑒 → (𝑔𝐶) = (𝑒𝐶))
3534eqeq2d 2770 . . . . . . . . 9 (𝑔 = 𝑒 → (𝑥 = (𝑔𝐶) ↔ 𝑥 = (𝑒𝐶)))
3635cbvrexv 3311 . . . . . . . 8 (∃𝑔𝐴 𝑥 = (𝑔𝐶) ↔ ∃𝑒𝐴 𝑥 = (𝑒𝐶))
37 sorpssi 7108 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑒𝑔𝑔𝑒))
38 ssdif 3888 . . . . . . . . . . . . . . . . 17 (𝑒𝑔 → (𝑒𝐶) ⊆ (𝑔𝐶))
39 ssdif 3888 . . . . . . . . . . . . . . . . 17 (𝑔𝑒 → (𝑔𝐶) ⊆ (𝑒𝐶))
4038, 39orim12i 539 . . . . . . . . . . . . . . . 16 ((𝑒𝑔𝑔𝑒) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
4137, 40syl 17 . . . . . . . . . . . . . . 15 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶)))
42 sseq2 3768 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓 ↔ (𝑒𝐶) ⊆ (𝑔𝐶)))
43 sseq1 3767 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑔𝐶) → (𝑓 ⊆ (𝑒𝐶) ↔ (𝑔𝐶) ⊆ (𝑒𝐶)))
4442, 43orbi12d 748 . . . . . . . . . . . . . . 15 (𝑓 = (𝑔𝐶) → (((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)) ↔ ((𝑒𝐶) ⊆ (𝑔𝐶) ∨ (𝑔𝐶) ⊆ (𝑒𝐶))))
4541, 44syl5ibrcom 237 . . . . . . . . . . . . . 14 (( [] Or 𝐴 ∧ (𝑒𝐴𝑔𝐴)) → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4645expr 644 . . . . . . . . . . . . 13 (( [] Or 𝐴𝑒𝐴) → (𝑔𝐴 → (𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))))
4746rexlimdv 3168 . . . . . . . . . . . 12 (( [] Or 𝐴𝑒𝐴) → (∃𝑔𝐴 𝑓 = (𝑔𝐶) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4812, 47syl5bi 232 . . . . . . . . . . 11 (( [] Or 𝐴𝑒𝐴) → (𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
4948ralrimiv 3103 . . . . . . . . . 10 (( [] Or 𝐴𝑒𝐴) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶)))
50 sseq1 3767 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑥𝑓 ↔ (𝑒𝐶) ⊆ 𝑓))
51 sseq2 3768 . . . . . . . . . . . 12 (𝑥 = (𝑒𝐶) → (𝑓𝑥𝑓 ⊆ (𝑒𝐶)))
5250, 51orbi12d 748 . . . . . . . . . . 11 (𝑥 = (𝑒𝐶) → ((𝑥𝑓𝑓𝑥) ↔ ((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5352ralbidv 3124 . . . . . . . . . 10 (𝑥 = (𝑒𝐶) → (∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥) ↔ ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))((𝑒𝐶) ⊆ 𝑓𝑓 ⊆ (𝑒𝐶))))
5449, 53syl5ibrcom 237 . . . . . . . . 9 (( [] Or 𝐴𝑒𝐴) → (𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5554rexlimdva 3169 . . . . . . . 8 ( [] Or 𝐴 → (∃𝑒𝐴 𝑥 = (𝑒𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5636, 55syl5bi 232 . . . . . . 7 ( [] Or 𝐴 → (∃𝑔𝐴 𝑥 = (𝑔𝐶) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5733, 56syl5bi 232 . . . . . 6 ( [] Or 𝐴 → (𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥)))
5857ralrimiv 3103 . . . . 5 ( [] Or 𝐴 → ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
59 sorpss 7107 . . . . 5 ( [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∀𝑥 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))∀𝑓 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))(𝑥𝑓𝑓𝑥))
6058, 59sylibr 224 . . . 4 ( [] Or 𝐴 → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
6130, 60syl 17 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))
62 fin2i 9309 . . 3 ((((𝐵𝐶) ∈ FinII ∧ ran (𝑔𝐴 ↦ (𝑔𝐶)) ⊆ 𝒫 (𝐵𝐶)) ∧ (ran (𝑔𝐴 ↦ (𝑔𝐶)) ≠ ∅ ∧ [] Or ran (𝑔𝐴 ↦ (𝑔𝐶)))) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
631, 16, 29, 61, 62syl22anc 1478 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
64 simpll3 1259 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ 𝐴𝐴)
65 difeq1 3864 . . . . . . 7 (𝑔 = 𝑓 → (𝑔𝐶) = (𝑓𝐶))
6665cbvmptv 4902 . . . . . 6 (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐴 ↦ (𝑓𝐶))
6766elrnmpt 5527 . . . . 5 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)))
6867ibi 256 . . . 4 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → ∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
69 eqid 2760 . . . . . . . . . . . . . . . 16 (𝐶) = (𝐶)
70 difeq1 3864 . . . . . . . . . . . . . . . . . 18 (𝑔 = → (𝑔𝐶) = (𝐶))
7170eqeq2d 2770 . . . . . . . . . . . . . . . . 17 (𝑔 = → ((𝐶) = (𝑔𝐶) ↔ (𝐶) = (𝐶)))
7271rspcev 3449 . . . . . . . . . . . . . . . 16 ((𝐴 ∧ (𝐶) = (𝐶)) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7369, 72mpan2 709 . . . . . . . . . . . . . . 15 (𝐴 → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7473adantl 473 . . . . . . . . . . . . . 14 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
75 vex 3343 . . . . . . . . . . . . . . 15 ∈ V
76 difexg 4960 . . . . . . . . . . . . . . 15 ( ∈ V → (𝐶) ∈ V)
7710elrnmpt 5527 . . . . . . . . . . . . . . 15 ((𝐶) ∈ V → ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶)))
7875, 76, 77mp2b 10 . . . . . . . . . . . . . 14 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝐶) = (𝑔𝐶))
7974, 78sylibr 224 . . . . . . . . . . . . 13 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
80 elssuni 4619 . . . . . . . . . . . . 13 ((𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
8179, 80syl 17 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ ran (𝑔𝐴 ↦ (𝑔𝐶)))
82 simplr 809 . . . . . . . . . . . 12 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
8381, 82sseqtrd 3782 . . . . . . . . . . 11 (((𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
8483adantll 752 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶) ⊆ (𝑓𝐶))
85 unss2 3927 . . . . . . . . . . 11 ((𝐶) ⊆ (𝑓𝐶) → (𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)))
86 uncom 3900 . . . . . . . . . . . . . . 15 (𝐶 ∪ (𝐶)) = ((𝐶) ∪ 𝐶)
87 undif1 4187 . . . . . . . . . . . . . . 15 ((𝐶) ∪ 𝐶) = (𝐶)
8886, 87eqtri 2782 . . . . . . . . . . . . . 14 (𝐶 ∪ (𝐶)) = (𝐶)
8988a1i 11 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝐶)) = (𝐶))
9064ad2antrr 764 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝐴𝐴)
9117ad2antrr 764 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝐴)
92 simplrr 820 . . . . . . . . . . . . . . . . 17 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
93 eqeq1 2764 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑒 = (𝑥𝐶) → (𝑒 = ∅ ↔ (𝑥𝐶) = ∅))
94 simpllr 817 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))
95 ssdif0 4085 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝐶 ↔ (𝑓𝐶) = ∅)
9695biimpi 206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝐶 → (𝑓𝐶) = ∅)
9796ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑓𝐶) = ∅)
9894, 97eqtrd 2794 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅)
99 uni0c 4616 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( ran (𝑔𝐴 ↦ (𝑔𝐶)) = ∅ ↔ ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
10098, 99sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → ∀𝑒 ∈ ran (𝑔𝐴 ↦ (𝑔𝐶))𝑒 = ∅)
101 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝐶) = (𝑥𝐶)
102 difeq1 3864 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑔 = 𝑥 → (𝑔𝐶) = (𝑥𝐶))
103102eqeq2d 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑔 = 𝑥 → ((𝑥𝐶) = (𝑔𝐶) ↔ (𝑥𝐶) = (𝑥𝐶)))
104103rspcev 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐴 ∧ (𝑥𝐶) = (𝑥𝐶)) → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
105101, 104mpan2 709 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝐴 → ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
106 difexg 4960 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ V → (𝑥𝐶) ∈ V)
10710elrnmpt 5527 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥𝐶) ∈ V → ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶)))
10831, 106, 107mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) ↔ ∃𝑔𝐴 (𝑥𝐶) = (𝑔𝐶))
109105, 108sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝐴 → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
110109adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
11193, 100, 110rspcdva 3455 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → (𝑥𝐶) = ∅)
112 ssdif0 4085 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥𝐶 ↔ (𝑥𝐶) = ∅)
113111, 112sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) ∧ 𝑥𝐴) → 𝑥𝐶)
114113ralrimiva 3104 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → ∀𝑥𝐴 𝑥𝐶)
115 unissb 4621 . . . . . . . . . . . . . . . . . . . . 21 ( 𝐴𝐶 ↔ ∀𝑥𝐴 𝑥𝐶)
116114, 115sylibr 224 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐶)
117 elssuni 4619 . . . . . . . . . . . . . . . . . . . . 21 (𝐶𝐴𝐶 𝐴)
118117ad2antrr 764 . . . . . . . . . . . . . . . . . . . 20 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶 𝐴)
119116, 118eqssd 3761 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴 = 𝐶)
120 simpll 807 . . . . . . . . . . . . . . . . . . 19 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐶𝐴)
121119, 120eqeltrd 2839 . . . . . . . . . . . . . . . . . 18 (((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) ∧ 𝑓𝐶) → 𝐴𝐴)
122121ex 449 . . . . . . . . . . . . . . . . 17 ((𝐶𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶)) → (𝑓𝐶 𝐴𝐴))
12391, 92, 122syl2anc 696 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶 𝐴𝐴))
12490, 123mtod 189 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ¬ 𝑓𝐶)
12530ad2antrr 764 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → [] Or 𝐴)
126 simplrl 819 . . . . . . . . . . . . . . . 16 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓𝐴)
127 sorpssi 7108 . . . . . . . . . . . . . . . 16 (( [] Or 𝐴 ∧ (𝑓𝐴𝐶𝐴)) → (𝑓𝐶𝐶𝑓))
128125, 126, 91, 127syl12anc 1475 . . . . . . . . . . . . . . 15 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝑓𝐶𝐶𝑓))
129 orel1 396 . . . . . . . . . . . . . . 15 𝑓𝐶 → ((𝑓𝐶𝐶𝑓) → 𝐶𝑓))
130124, 128, 129sylc 65 . . . . . . . . . . . . . 14 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝐶𝑓)
131 undif 4193 . . . . . . . . . . . . . 14 (𝐶𝑓 ↔ (𝐶 ∪ (𝑓𝐶)) = 𝑓)
132130, 131sylib 208 . . . . . . . . . . . . 13 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → (𝐶 ∪ (𝑓𝐶)) = 𝑓)
13389, 132sseq12d 3775 . . . . . . . . . . . 12 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) ↔ (𝐶) ⊆ 𝑓))
134 ssun1 3919 . . . . . . . . . . . . 13 ⊆ (𝐶)
135 sstr 3752 . . . . . . . . . . . . 13 (( ⊆ (𝐶) ∧ (𝐶) ⊆ 𝑓) → 𝑓)
136134, 135mpan 708 . . . . . . . . . . . 12 ((𝐶) ⊆ 𝑓𝑓)
137133, 136syl6bi 243 . . . . . . . . . . 11 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶 ∪ (𝐶)) ⊆ (𝐶 ∪ (𝑓𝐶)) → 𝑓))
13885, 137syl5 34 . . . . . . . . . 10 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → ((𝐶) ⊆ (𝑓𝐶) → 𝑓))
13984, 138mpd 15 . . . . . . . . 9 ((((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) ∧ 𝐴) → 𝑓)
140139ralrimiva 3104 . . . . . . . 8 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → ∀𝐴 𝑓)
141 unissb 4621 . . . . . . . 8 ( 𝐴𝑓 ↔ ∀𝐴 𝑓)
142140, 141sylibr 224 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝑓)
143 elssuni 4619 . . . . . . . 8 (𝑓𝐴𝑓 𝐴)
144143ad2antrl 766 . . . . . . 7 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓 𝐴)
145142, 144eqssd 3761 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴 = 𝑓)
146 simprl 811 . . . . . 6 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝑓𝐴)
147145, 146eqeltrd 2839 . . . . 5 (((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) ∧ (𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶))) → 𝐴𝐴)
148147rexlimdvaa 3170 . . . 4 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → (∃𝑓𝐴 ran (𝑔𝐴 ↦ (𝑔𝐶)) = (𝑓𝐶) → 𝐴𝐴))
14968, 148syl5 34 . . 3 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ( ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)) → 𝐴𝐴))
15064, 149mtod 189 . 2 ((((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) ∧ (𝐵𝐶) ∈ FinII) → ¬ ran (𝑔𝐴 ↦ (𝑔𝐶)) ∈ ran (𝑔𝐴 ↦ (𝑔𝐶)))
15163, 150pm2.65da 601 1 (((𝐴 ⊆ 𝒫 𝐵 ∧ [] Or 𝐴 ∧ ¬ 𝐴𝐴) ∧ (¬ 𝐶 ∈ Fin ∧ 𝐶𝐴)) → ¬ (𝐵𝐶) ∈ FinII)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ∃wrex 3051  Vcvv 3340   ∖ cdif 3712   ∪ cun 3713   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  ∪ cuni 4588   ↦ cmpt 4881   Or wor 5186  ran crn 5267   [⊊] crpss 7101  Fincfn 8121  FinIIcfin2 9293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-rpss 7102  df-fin2 9300 This theorem is referenced by:  fin1a2s  9428
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