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Theorem fompt 39195
Description: Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
fompt.1 𝐹 = (𝑥𝐴𝐶)
Assertion
Ref Expression
fompt (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑦,𝐹
Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fompt
StepHypRef Expression
1 fompt.1 . . . . . . 7 𝐹 = (𝑥𝐴𝐶)
2 nfmpt1 4738 . . . . . . 7 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2760 . . . . . 6 𝑥𝐹
43dffo3f 39180 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
54simplbi 476 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
61fmpt 6367 . . . . . 6 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76bicomi 214 . . . . 5 (𝐹:𝐴𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
87biimpi 206 . . . 4 (𝐹:𝐴𝐵 → ∀𝑥𝐴 𝐶𝐵)
95, 8syl 17 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴 𝐶𝐵)
103foelrnf 39189 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
11 nfcv 2762 . . . . . . . 8 𝑥𝐴
12 nfcv 2762 . . . . . . . 8 𝑥𝐵
133, 11, 12nffo 6101 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
14 simpr 477 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
15 simpr 477 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
169r19.21bi 2929 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
171fvmpt2 6278 . . . . . . . . . . . 12 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1815, 16, 17syl2anc 692 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1918adantr 481 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
2014, 19eqtrd 2654 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = 𝐶)
2120ex 450 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶))
2221ex 450 . . . . . . 7 (𝐹:𝐴onto𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶)))
2313, 22reximdai 3009 . . . . . 6 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2423adantr 481 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2510, 24mpd 15 . . . 4 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
2625ralrimiva 2963 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
279, 26jca 554 . 2 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
286biimpi 206 . . . . 5 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2928adantr 481 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴𝐵)
30 nfv 1841 . . . . . 6 𝑦𝑥𝐴 𝐶𝐵
31 nfra1 2938 . . . . . 6 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
3230, 31nfan 1826 . . . . 5 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
33 simpll 789 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
34 rspa 2927 . . . . . . . 8 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3534adantll 749 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
36 nfra1 2938 . . . . . . . . 9 𝑥𝑥𝐴 𝐶𝐵
37 simp3 1061 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
38 simpr 477 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
39 rspa 2927 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
4038, 39, 17syl2anc 692 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
4140eqcomd 2626 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
42413adant3 1079 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
4337, 42eqtrd 2654 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
44433exp 1262 . . . . . . . . 9 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4536, 44reximdai 3009 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4645imp 445 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4733, 35, 46syl2anc 692 . . . . . 6 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4847ex 450 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4932, 48ralrimi 2954 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
5029, 49jca 554 . . 3 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
5150, 4sylibr 224 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴onto𝐵)
5227, 51impbii 199 1 (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910  cmpt 4720  wf 5872  ontowfo 5874  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884
This theorem is referenced by:  disjinfi  39196
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