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Theorem f1o3d 29740
Description: Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
Hypotheses
Ref Expression
f1o3d.1 (𝜑𝐹 = (𝑥𝐴𝐶))
f1o3d.2 ((𝜑𝑥𝐴) → 𝐶𝐵)
f1o3d.3 ((𝜑𝑦𝐵) → 𝐷𝐴)
f1o3d.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
Assertion
Ref Expression
f1o3d (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem f1o3d
StepHypRef Expression
1 f1o3d.2 . . . . . 6 ((𝜑𝑥𝐴) → 𝐶𝐵)
21ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
3 eqid 2760 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43fnmpt 6181 . . . . 5 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴𝐶) Fn 𝐴)
52, 4syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
6 f1o3d.1 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐶))
76fneq1d 6142 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
85, 7mpbird 247 . . 3 (𝜑𝐹 Fn 𝐴)
9 f1o3d.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝐴)
109ralrimiva 3104 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝐴)
11 eqid 2760 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
1211fnmpt 6181 . . . . 5 (∀𝑦𝐵 𝐷𝐴 → (𝑦𝐵𝐷) Fn 𝐵)
1310, 12syl 17 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
14 eleq1a 2834 . . . . . . . . . . 11 (𝐶𝐵 → (𝑦 = 𝐶𝑦𝐵))
151, 14syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑦 = 𝐶𝑦𝐵))
1615impr 650 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → 𝑦𝐵)
17 f1o3d.4 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
1817biimpar 503 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
1918exp42 640 . . . . . . . . . . 11 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑦 = 𝐶𝑥 = 𝐷))))
2019com34 91 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → (𝑦𝐵𝑥 = 𝐷))))
2120imp32 448 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
2216, 21jcai 560 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
23 eleq1a 2834 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 = 𝐷𝑥𝐴))
249, 23syl 17 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑥 = 𝐷𝑥𝐴))
2524impr 650 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → 𝑥𝐴)
2617biimpa 502 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
2726exp42 640 . . . . . . . . . . . 12 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑥 = 𝐷𝑦 = 𝐶))))
2827com23 86 . . . . . . . . . . 11 (𝜑 → (𝑦𝐵 → (𝑥𝐴 → (𝑥 = 𝐷𝑦 = 𝐶))))
2928com34 91 . . . . . . . . . 10 (𝜑 → (𝑦𝐵 → (𝑥 = 𝐷 → (𝑥𝐴𝑦 = 𝐶))))
3029imp32 448 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
3125, 30jcai 560 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
3222, 31impbida 913 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
3332opabbidv 4868 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
34 df-mpt 4882 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
356, 34syl6eq 2810 . . . . . . . 8 (𝜑𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
3635cnveqd 5453 . . . . . . 7 (𝜑𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
37 cnvopab 5691 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3836, 37syl6eq 2810 . . . . . 6 (𝜑𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
39 df-mpt 4882 . . . . . . 7 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
4039a1i 11 . . . . . 6 (𝜑 → (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
4133, 38, 403eqtr4d 2804 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
4241fneq1d 6142 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
4313, 42mpbird 247 . . 3 (𝜑𝐹 Fn 𝐵)
44 dff1o4 6306 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
458, 43, 44sylanbrc 701 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
4645, 41jca 555 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {copab 4864  cmpt 4881  ccnv 5265   Fn wfn 6044  1-1-ontowf1o 6048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056
This theorem is referenced by:  fmptco1f1o  29743  ballotlemsf1o  30884
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