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Theorem fompt 42958
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 5195 . . . . . . 7 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2903 . . . . . 6 𝑥𝐹
43dffo3f 42945 . . . . 5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
54simplbi 498 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
61fmpt 7023 . . . . . 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 42952 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
11 nfcv 2905 . . . . . . . 8 𝑥𝐴
12 nfcv 2905 . . . . . . . 8 𝑥𝐵
133, 11, 12nffo 6724 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
14 simpr 485 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
15 simpr 485 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
169r19.21bi 3231 . . . . . . . . . . . 12 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
171fvmpt2 6925 . . . . . . . . . . . 12 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1815, 16, 17syl2anc 584 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1918adantr 481 . . . . . . . . . 10 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
2014, 19eqtrd 2777 . . . . . . . . 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 3241 . . . . . 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 3140 . . 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 1916 . . . . . 6 𝑦𝑥𝐴 𝐶𝐵
31 nfra1 3264 . . . . . 6 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
3230, 31nfan 1901 . . . . 5 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
33 simpll 764 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
34 rspa 3228 . . . . . . . 8 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3534adantll 711 . . . . . . 7 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
36 nfra1 3264 . . . . . . . . 9 𝑥𝑥𝐴 𝐶𝐵
37 simp3 1137 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
38 simpr 485 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
39 rspa 3228 . . . . . . . . . . . . . 14 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
4038, 39, 17syl2anc 584 . . . . . . . . . . . . 13 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
4140eqcomd 2743 . . . . . . . . . . . 12 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
42413adant3 1131 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
4337, 42eqtrd 2777 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
44433exp 1118 . . . . . . . . 9 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4536, 44reximdai 3241 . . . . . . . 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 3237 . . . 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 1086   = wceq 1540  wcel 2105  wral 3062  wrex 3071  cmpt 5170  wf 6461  ontowfo 6463  cfv 6465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fo 6471  df-fv 6473
This theorem is referenced by:  disjinfi  42959
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