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Theorem dffo3f 42255
Description: An onto mapping expressed in terms of function values. As dffo3 6878 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 6596 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵))
2 ffn 6504 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
3 fnrnfv 6729 . . . . . . 7 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)})
4 dffo3f.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2899 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6684 . . . . . . . . . 10 𝑥(𝐹𝑤)
76nfeq2 2916 . . . . . . . . 9 𝑥 𝑦 = (𝐹𝑤)
8 nfv 1921 . . . . . . . . 9 𝑤 𝑦 = (𝐹𝑥)
9 fveq2 6674 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
109eqeq2d 2749 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑦 = (𝐹𝑤) ↔ 𝑦 = (𝐹𝑥)))
117, 8, 10cbvrexw 3341 . . . . . . . 8 (∃𝑤𝐴 𝑦 = (𝐹𝑤) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
1211abbii 2803 . . . . . . 7 {𝑦 ∣ ∃𝑤𝐴 𝑦 = (𝐹𝑤)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
133, 12eqtrdi 2789 . . . . . 6 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
1413eqeq1d 2740 . . . . 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 dfbi2 478 . . . . . . 7 ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
17 nfcv 2899 . . . . . . . . . 10 𝑥𝐴
18 nfcv 2899 . . . . . . . . . 10 𝑥𝐵
194, 17, 18nff 6500 . . . . . . . . 9 𝑥 𝐹:𝐴𝐵
20 nfv 1921 . . . . . . . . 9 𝑥 𝑦𝐵
21 simpr 488 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦 = (𝐹𝑥))
22 ffvelrn 6859 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2322adantr 484 . . . . . . . . . 10 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → (𝐹𝑥) ∈ 𝐵)
2421, 23eqeltrd 2833 . . . . . . . . 9 (((𝐹:𝐴𝐵𝑥𝐴) ∧ 𝑦 = (𝐹𝑥)) → 𝑦𝐵)
2519, 20, 24rexlimd3 42231 . . . . . . . 8 (𝐹:𝐴𝐵 → (∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵))
2625biantrurd 536 . . . . . . 7 (𝐹:𝐴𝐵 → ((𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)) ↔ ((∃𝑥𝐴 𝑦 = (𝐹𝑥) → 𝑦𝐵) ∧ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))))
2716, 26bitr4id 293 . . . . . 6 (𝐹:𝐴𝐵 → ((∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ (𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
2827albidv 1927 . . . . 5 (𝐹:𝐴𝐵 → (∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥))))
29 abeq1 2865 . . . . 5 ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦(∃𝑥𝐴 𝑦 = (𝐹𝑥) ↔ 𝑦𝐵))
30 df-ral 3058 . . . . 5 (∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥) ↔ ∀𝑦(𝑦𝐵 → ∃𝑥𝐴 𝑦 = (𝐹𝑥)))
3128, 29, 303bitr4g 317 . . . 4 (𝐹:𝐴𝐵 → ({𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)} = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3215, 31bitrd 282 . . 3 (𝐹:𝐴𝐵 → (ran 𝐹 = 𝐵 ↔ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
3332pm5.32i 578 . 2 ((𝐹:𝐴𝐵 ∧ ran 𝐹 = 𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
341, 33bitri 278 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑦 = (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wcel 2114  {cab 2716  wnfc 2879  wral 3053  wrex 3054  ran crn 5526   Fn wfn 6334  wf 6335  ontowfo 6337  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347
This theorem is referenced by:  foelrnf  42262  fompt  42268
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