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Theorem fompt 39899
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 4881 . . . . . . 7 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2911 . . . . . 6 𝑥𝐹
43dffo3f 39884 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
54simplbi 485 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
61fmpt 6523 . . . . . 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 39893 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
11 nfcv 2913 . . . . . . . 8 𝑥𝐴
12 nfcv 2913 . . . . . . . 8 𝑥𝐵
133, 11, 12nffo 6255 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
14 simpr 471 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
15 simpr 471 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
169r19.21bi 3081 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
171fvmpt2 6433 . . . . . . . . . . . 12 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1815, 16, 17syl2anc 573 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1918adantr 466 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
2014, 19eqtrd 2805 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = 𝐶)
2120ex 397 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶))
2221ex 397 . . . . . . 7 (𝐹:𝐴onto𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶)))
2313, 22reximdai 3160 . . . . . 6 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2423adantr 466 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2510, 24mpd 15 . . . 4 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
2625ralrimiva 3115 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
279, 26jca 501 . 2 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
286biimpi 206 . . . . 5 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2928adantr 466 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴𝐵)
30 nfv 1995 . . . . . 6 𝑦𝑥𝐴 𝐶𝐵
31 nfra1 3090 . . . . . 6 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
3230, 31nfan 1980 . . . . 5 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
33 simpll 750 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
34 rspa 3079 . . . . . . . 8 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3534adantll 693 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
36 nfra1 3090 . . . . . . . . 9 𝑥𝑥𝐴 𝐶𝐵
37 simp3 1132 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
38 simpr 471 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
39 rspa 3079 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
4038, 39, 17syl2anc 573 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
4140eqcomd 2777 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
42413adant3 1126 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
4337, 42eqtrd 2805 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
44433exp 1112 . . . . . . . . 9 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4536, 44reximdai 3160 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4645imp 393 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4733, 35, 46syl2anc 573 . . . . . 6 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4847ex 397 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4932, 48ralrimi 3106 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
5029, 49jca 501 . . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cmpt 4863  wf 6027  ontowfo 6029  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039
This theorem is referenced by:  disjinfi  39900
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