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Theorem dffo3f 41305
Description: An onto mapping expressed in terms of function values. As dffo3 6863 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
dffo3f.1 𝑥𝐹
Assertion
Ref Expression
dffo3f (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐹
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffo3f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dffo2 6590 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 6510 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 6721 . . . . . . 7 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)})
4 dffo3f.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2981 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6676 . . . . . . . . . 10 𝑥(𝐹𝑤)
76nfeq2 2999 . . . . . . . . 9 𝑥 𝑦 = (𝐹𝑤)
8 nfv 1908 . . . . . . . . 9 𝑤 𝑦 = (𝐹𝑥)
9 fveq2 6666 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
109eqeq2d 2836 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦 = (𝐹𝑤) ↔ 𝑦 = (𝐹𝑥)))
117, 8, 10cbvrex 3451 . . . . . . . 8 (∃𝑤𝐴 𝑦 = (𝐹𝑤) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1211abbii 2890 . . . . . . 7 {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
133, 12syl6eq 2876 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1413eqeq1d 2827 . . . . 5 (𝐹 Fn 𝐴 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
152, 14syl 17 . . . 4 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵))
16 nfcv 2981 . . . . . . . . . 10 𝑥𝐴
17 nfcv 2981 . . . . . . . . . 10 𝑥𝐵
184, 16, 17nff 6506 . . . . . . . . 9 𝑥 𝐹:𝐴𝐵
19 nfv 1908 . . . . . . . . 9 𝑥 𝑦𝐵
20 simpr 485 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
21 ffvelrn 6844 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2221adantr 481 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
2320, 22eqeltrd 2917 . . . . . . . . 9 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
2418, 19, 23rexlimd3 41280 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
2524biantrurd 533 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
26 dfbi2 475 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
2725, 26syl6rbbr 291 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
2827albidv 1914 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
29 abeq1 2950 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
30 df-ral 3147 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
3128, 29, 303bitr4g 315 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3215, 31bitrd 280 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3332pm5.32i 575 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
341, 33bitri 276 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wcel 2107  {cab 2803  wnfc 2965  wral 3142  wrex 3143  ran crn 5554   Fn wfn 6346  wf 6347  ontowfo 6349  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359
This theorem is referenced by:  foelrnf  41314  fompt  41320
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