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Theorem fompt 43401
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 5213 . . . . . . 7 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2905 . . . . . 6 𝑥𝐹
43dffo3f 43388 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
54simplbi 498 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
61fmpt 7058 . . . . . 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 43395 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
11 nfcv 2907 . . . . . . . 8 𝑥𝐴
12 nfcv 2907 . . . . . . . 8 𝑥𝐵
133, 11, 12nffo 6755 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
14 simpr 485 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
15 simpr 485 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
169r19.21bi 3234 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
171fvmpt2 6959 . . . . . . . . . . . 12 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1815, 16, 17syl2anc 584 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1918adantr 481 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
2014, 19eqtrd 2776 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = 𝐶)
2120ex 413 . . . . . . . 8 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶))
2221ex 413 . . . . . . 7 (𝐹:𝐴onto𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶)))
2313, 22reximdai 3244 . . . . . 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 3143 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
279, 26jca 512 . 2 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
286biimpi 215 . . . . 5 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2928adantr 481 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴𝐵)
30 nfv 1917 . . . . . 6 𝑦𝑥𝐴 𝐶𝐵
31 nfra1 3267 . . . . . 6 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
3230, 31nfan 1902 . . . . 5 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
33 simpll 765 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
34 rspa 3231 . . . . . . . 8 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3534adantll 712 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
36 nfra1 3267 . . . . . . . . 9 𝑥𝑥𝐴 𝐶𝐵
37 simp3 1138 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
38 simpr 485 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
39 rspa 3231 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
4038, 39, 17syl2anc 584 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
4140eqcomd 2742 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
42413adant3 1132 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
4337, 42eqtrd 2776 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
44433exp 1119 . . . . . . . . 9 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4536, 44reximdai 3244 . . . . . . . 8 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4645imp 407 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4733, 35, 46syl2anc 584 . . . . . 6 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4847ex 413 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4932, 48ralrimi 3240 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
5029, 49jca 512 . . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  cmpt 5188  wf 6492  ontowfo 6494  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504
This theorem is referenced by:  disjinfi  43402
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