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Theorem fompt 43890
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 5257 . . . . . . 7 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2902 . . . . . 6 𝑥𝐹
43dffo3f 43877 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
54simplbi 499 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
61fmpt 7110 . . . . . 6 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76bicomi 223 . . . . 5 (𝐹:𝐴𝐵 ↔ ∀𝑥𝐴 𝐶𝐵)
87biimpi 215 . . . 4 (𝐹:𝐴𝐵 → ∀𝑥𝐴 𝐶𝐵)
95, 8syl 17 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴 𝐶𝐵)
103foelrnf 43884 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
11 nfcv 2904 . . . . . . . 8 𝑥𝐴
12 nfcv 2904 . . . . . . . 8 𝑥𝐵
133, 11, 12nffo 6805 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
14 simpr 486 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
15 simpr 486 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
169r19.21bi 3249 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
171fvmpt2 7010 . . . . . . . . . . . 12 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1815, 16, 17syl2anc 585 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1918adantr 482 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
2014, 19eqtrd 2773 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = 𝐶)
2120ex 414 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶))
2221ex 414 . . . . . . 7 (𝐹:𝐴onto𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶)))
2313, 22reximdai 3259 . . . . . 6 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2423adantr 482 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2510, 24mpd 15 . . . 4 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
2625ralrimiva 3147 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
279, 26jca 513 . 2 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
286biimpi 215 . . . . 5 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2928adantr 482 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴𝐵)
30 nfv 1918 . . . . . 6 𝑦𝑥𝐴 𝐶𝐵
31 nfra1 3282 . . . . . 6 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
3230, 31nfan 1903 . . . . 5 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
33 simpll 766 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
34 rspa 3246 . . . . . . . 8 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3534adantll 713 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
36 nfra1 3282 . . . . . . . . 9 𝑥𝑥𝐴 𝐶𝐵
37 simp3 1139 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
38 simpr 486 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
39 rspa 3246 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
4038, 39, 17syl2anc 585 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
4140eqcomd 2739 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
42413adant3 1133 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
4337, 42eqtrd 2773 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
44433exp 1120 . . . . . . . . 9 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4536, 44reximdai 3259 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4645imp 408 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4733, 35, 46syl2anc 585 . . . . . 6 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4847ex 414 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4932, 48ralrimi 3255 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
5029, 49jca 513 . . 3 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
5150, 4sylibr 233 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴onto𝐵)
5227, 51impbii 208 1 (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cmpt 5232  wf 6540  ontowfo 6542  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552
This theorem is referenced by:  disjinfi  43891
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