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Theorem resf1o 29735
 Description: Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
Hypotheses
Ref Expression
resf1o.1 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
resf1o.2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
Assertion
Ref Expression
resf1o (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑍
Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem resf1o
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resf1o.2 . 2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
2 resexg 5552 . . 3 (𝑓𝑋 → (𝑓𝐶) ∈ V)
32adantl 473 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑓𝑋) → (𝑓𝐶) ∈ V)
4 simpr 479 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → 𝑔 ∈ (𝐵𝑚 𝐶))
5 difexg 4916 . . . . . . 7 (𝐴𝑉 → (𝐴𝐶) ∈ V)
653ad2ant1 1125 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝐴) → (𝐴𝐶) ∈ V)
7 snex 5013 . . . . . 6 {𝑍} ∈ V
8 xpexg 7077 . . . . . 6 (((𝐴𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴𝐶) × {𝑍}) ∈ V)
96, 7, 8sylancl 697 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝐴𝐶) × {𝑍}) ∈ V)
109adantr 472 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → ((𝐴𝐶) × {𝑍}) ∈ V)
11 unexg 7076 . . . 4 ((𝑔 ∈ (𝐵𝑚 𝐶) ∧ ((𝐴𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
124, 10, 11syl2anc 696 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
1312adantlr 753 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
14 resf1o.1 . . . . 5 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
1514rabeq2i 3301 . . . 4 (𝑓𝑋 ↔ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
1615anbi1i 733 . . 3 ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
17 simprr 813 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 = (𝑓𝐶))
18 simprll 821 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 ∈ (𝐵𝑚 𝐴))
19 elmapi 7996 . . . . . . . . 9 (𝑓 ∈ (𝐵𝑚 𝐴) → 𝑓:𝐴𝐵)
2018, 19syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓:𝐴𝐵)
21 simp3 1130 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶𝐴)
2221ad2antrr 764 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐶𝐴)
2320, 22fssresd 6184 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶):𝐶𝐵)
24 simp2 1129 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐵𝑊)
25 simp1 1128 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐴𝑉)
2625, 21ssexd 4913 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶 ∈ V)
27 elmapg 7987 . . . . . . . . 9 ((𝐵𝑊𝐶 ∈ V) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2824, 26, 27syl2anc 696 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2928ad2antrr 764 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
3023, 29mpbird 247 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) ∈ (𝐵𝑚 𝐶))
3117, 30eqeltrd 2803 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 ∈ (𝐵𝑚 𝐶))
32 undif 4157 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3332biimpi 206 . . . . . . . . . 10 (𝐶𝐴 → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3433reseq2d 5503 . . . . . . . . 9 (𝐶𝐴 → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
3522, 34syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
36 ffn 6158 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
37 fnresdm 6113 . . . . . . . . 9 (𝑓 Fn 𝐴 → (𝑓𝐴) = 𝑓)
3820, 36, 373syl 18 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐴) = 𝑓)
3935, 38eqtr2d 2759 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))))
40 resundi 5520 . . . . . . 7 (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶)))
4139, 40syl6eq 2774 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))))
4217eqcomd 2730 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) = 𝑔)
43 simprlr 822 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
4425ad2antrr 764 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐴𝑉)
45 simplr 809 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑍𝐵)
46 eqid 2724 . . . . . . . . . . 11 (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
4746ffs2 29733 . . . . . . . . . 10 ((𝐴𝑉𝑍𝐵𝑓:𝐴𝐵) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
4844, 45, 20, 47syl3anc 1439 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
49 sseqin2 3925 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
5049biimpi 206 . . . . . . . . . 10 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
5122, 50syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝐴𝐶) = 𝐶)
5243, 48, 513sstr4d 3754 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴𝐶))
53 simpl 474 . . . . . . . . . . . 12 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
5453, 19, 363syl 18 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn 𝐴)
55 inundif 4154 . . . . . . . . . . . 12 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5655fneq2i 6099 . . . . . . . . . . 11 (𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ↔ 𝑓 Fn 𝐴)
5754, 56sylibr 224 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)))
58 vex 3307 . . . . . . . . . . 11 𝑓 ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ V)
60 simpr 479 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑍𝐵)
61 inindif 29581 . . . . . . . . . . 11 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
6261a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
63 fnsuppres 7442 . . . . . . . . . 10 ((𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ∧ (𝑓 ∈ V ∧ 𝑍𝐵) ∧ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6457, 59, 60, 62, 63syl121anc 1444 . . . . . . . . 9 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6518, 45, 64syl2anc 696 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6652, 65mpbid 222 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍}))
6742, 66uneq12d 3876 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))) = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6841, 67eqtrd 2758 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6931, 68jca 555 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍}))))
7024ad2antrr 764 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐵𝑊)
7125ad2antrr 764 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐴𝑉)
72 elmapi 7996 . . . . . . . . 9 (𝑔 ∈ (𝐵𝑚 𝐶) → 𝑔:𝐶𝐵)
7372ad2antrl 766 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔:𝐶𝐵)
74 simplr 809 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑍𝐵)
75 fconst6g 6207 . . . . . . . . 9 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
7674, 75syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
77 disjdif 4148 . . . . . . . . 9 (𝐶 ∩ (𝐴𝐶)) = ∅
7877a1i 11 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴𝐶)) = ∅)
79 fun2 6180 . . . . . . . 8 (((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵) ∧ (𝐶 ∩ (𝐴𝐶)) = ∅) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
8073, 76, 78, 79syl21anc 1438 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
81 simprr 813 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
8281eqcomd 2730 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) = 𝑓)
8321ad2antrr 764 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐶𝐴)
8483, 33syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
8582, 84feq12d 6146 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵𝑓:𝐴𝐵))
8680, 85mpbid 222 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓:𝐴𝐵)
87 elmapg 7987 . . . . . . 7 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
8887biimpar 503 . . . . . 6 (((𝐵𝑊𝐴𝑉) ∧ 𝑓:𝐴𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
8970, 71, 86, 88syl21anc 1438 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 ∈ (𝐵𝑚 𝐴))
9071, 74, 86, 47syl3anc 1439 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
9181adantr 472 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
9291fveq1d 6306 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥))
9373adantr 472 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔:𝐶𝐵)
94 ffn 6158 . . . . . . . . . 10 (𝑔:𝐶𝐵𝑔 Fn 𝐶)
9593, 94syl 17 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔 Fn 𝐶)
96 fconstg 6205 . . . . . . . . . . 11 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9796ad3antlr 769 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
98 ffn 6158 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
9997, 98syl 17 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
10077a1i 11 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝐶 ∩ (𝐴𝐶)) = ∅)
101 simpr 479 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
102 fvun2 6384 . . . . . . . . 9 ((𝑔 Fn 𝐶 ∧ ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶) ∧ ((𝐶 ∩ (𝐴𝐶)) = ∅ ∧ 𝑥 ∈ (𝐴𝐶))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
10395, 99, 100, 101, 102syl112anc 1443 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
104 fvconst 6546 . . . . . . . . 9 ((((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10597, 101, 104syl2anc 696 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10692, 103, 1053eqtrd 2762 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = 𝑍)
10786, 106suppss 7445 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
10890, 107eqsstr3d 3746 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
10981reseq1d 5502 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓𝐶) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶))
110 res0 5507 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}) ↾ ∅) = ∅
111 res0 5507 . . . . . . . . . 10 (𝑔 ↾ ∅) = ∅
112110, 111eqtr4i 2749 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
11377reseq2i 5500 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ ∅)
11477reseq2i 5500 . . . . . . . . 9 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (𝑔 ↾ ∅)
115112, 113, 1143eqtr4ri 2757 . . . . . . . 8 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))
116115a1i 11 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))))
117 fresaunres1 6190 . . . . . . 7 ((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵 ∧ (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
11873, 76, 116, 117syl3anc 1439 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
119109, 118eqtr2d 2759 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔 = (𝑓𝐶))
12089, 108, 119jca31 558 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
12169, 120impbida 913 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → (((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
12216, 121syl5bb 272 . 2 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
1231, 3, 13, 122f1od 7002 1 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1596   ∈ wcel 2103  {crab 3018  Vcvv 3304   ∖ cdif 3677   ∪ cun 3678   ∩ cin 3679   ⊆ wss 3680  ∅c0 4023  {csn 4285   ↦ cmpt 4837   × cxp 5216  ◡ccnv 5217   ↾ cres 5220   “ cima 5221   Fn wfn 5996  ⟶wf 5997  –1-1-onto→wf1o 6000  ‘cfv 6001  (class class class)co 6765   supp csupp 7415   ↑𝑚 cmap 7974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-supp 7416  df-map 7976 This theorem is referenced by:  eulerpartgbij  30664
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