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Theorem fompt 7138
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 fof 6821 . . . 4 (𝐹:𝐴onto𝐵𝐹:𝐴𝐵)
2 fompt.1 . . . . 5 𝐹 = (𝑥𝐴𝐶)
32fmpt 7130 . . . 4 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
41, 3sylibr 234 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑥𝐴 𝐶𝐵)
5 nfmpt1 5256 . . . . . . 7 𝑥(𝑥𝐴𝐶)
62, 5nfcxfr 2901 . . . . . 6 𝑥𝐹
76foelrnf 7128 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
8 nfcv 2903 . . . . . . . 8 𝑥𝐴
9 nfcv 2903 . . . . . . . 8 𝑥𝐵
106, 8, 9nffo 6820 . . . . . . 7 𝑥 𝐹:𝐴onto𝐵
11 simpr 484 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
12 simpr 484 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝑥𝐴)
134r19.21bi 3249 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑥𝐴) → 𝐶𝐵)
142fvmpt2 7027 . . . . . . . . . . 11 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
1512, 13, 14syl2anc 584 . . . . . . . . . 10 ((𝐹:𝐴onto𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
1615adantr 480 . . . . . . . . 9 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) = 𝐶)
1711, 16eqtrd 2775 . . . . . . . 8 (((𝐹:𝐴onto𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = 𝐶)
1817exp31 419 . . . . . . 7 (𝐹:𝐴onto𝐵 → (𝑥𝐴 → (𝑦 = (𝐹𝑥) → 𝑦 = 𝐶)))
1910, 18reximdai 3259 . . . . . 6 (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
2019adantr 480 . . . . 5 ((𝐹:𝐴onto𝐵𝑦𝐵) → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → ∃𝑥𝐴 𝑦 = 𝐶))
217, 20mpd 15 . . . 4 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
2221ralrimiva 3144 . . 3 (𝐹:𝐴onto𝐵 → ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
234, 22jca 511 . 2 (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
243biimpi 216 . . . 4 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2524adantr 480 . . 3 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴𝐵)
26 nfv 1912 . . . . 5 𝑦𝑥𝐴 𝐶𝐵
27 nfra1 3282 . . . . 5 𝑦𝑦𝐵𝑥𝐴 𝑦 = 𝐶
2826, 27nfan 1897 . . . 4 𝑦(∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶)
29 simpll 767 . . . . 5 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∀𝑥𝐴 𝐶𝐵)
30 rspa 3246 . . . . . 6 ((∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
3130adantll 714 . . . . 5 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = 𝐶)
32 nfra1 3282 . . . . . 6 𝑥𝑥𝐴 𝐶𝐵
33 simp3 1137 . . . . . . . 8 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = 𝐶)
34 simpr 484 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝑥𝐴)
35 rspa 3246 . . . . . . . . . . 11 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶𝐵)
3634, 35, 14syl2anc 584 . . . . . . . . . 10 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → (𝐹𝑥) = 𝐶)
3736eqcomd 2741 . . . . . . . . 9 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴) → 𝐶 = (𝐹𝑥))
38373adant3 1131 . . . . . . . 8 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝐶 = (𝐹𝑥))
3933, 38eqtrd 2775 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵𝑥𝐴𝑦 = 𝐶) → 𝑦 = (𝐹𝑥))
40393exp 1118 . . . . . 6 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴 → (𝑦 = 𝐶𝑦 = (𝐹𝑥))))
4132, 40reximdai 3259 . . . . 5 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
4229, 31, 41sylc 65 . . . 4 (((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) ∧ 𝑦𝐵) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
4328, 42ralrimia 3256 . . 3 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥))
446dffo3f 7126 . . 3 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
4525, 43, 44sylanbrc 583 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶) → 𝐹:𝐴onto𝐵)
4623, 45impbii 209 1 (𝐹:𝐴onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cmpt 5231  wf 6559  ontowfo 6561  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by:  zringfrac  33562  algextdeglem8  33730  disjinfi  45135
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