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Theorem f1o3d 29764
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 3161 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
3 eqid 2813 . . . . . 6 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43fnmpt 6234 . . . . 5 (∀𝑥𝐴 𝐶𝐵 → (𝑥𝐴𝐶) Fn 𝐴)
52, 4syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶) Fn 𝐴)
6 f1o3d.1 . . . . 5 (𝜑𝐹 = (𝑥𝐴𝐶))
76fneq1d 6195 . . . 4 (𝜑 → (𝐹 Fn 𝐴 ↔ (𝑥𝐴𝐶) Fn 𝐴))
85, 7mpbird 248 . . 3 (𝜑𝐹 Fn 𝐴)
9 f1o3d.3 . . . . . 6 ((𝜑𝑦𝐵) → 𝐷𝐴)
109ralrimiva 3161 . . . . 5 (𝜑 → ∀𝑦𝐵 𝐷𝐴)
11 eqid 2813 . . . . . 6 (𝑦𝐵𝐷) = (𝑦𝐵𝐷)
1211fnmpt 6234 . . . . 5 (∀𝑦𝐵 𝐷𝐴 → (𝑦𝐵𝐷) Fn 𝐵)
1310, 12syl 17 . . . 4 (𝜑 → (𝑦𝐵𝐷) Fn 𝐵)
14 eleq1a 2887 . . . . . . . . . . 11 (𝐶𝐵 → (𝑦 = 𝐶𝑦𝐵))
151, 14syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑦 = 𝐶𝑦𝐵))
1615impr 444 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → 𝑦𝐵)
17 f1o3d.4 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))
1817biimpar 465 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑦 = 𝐶) → 𝑥 = 𝐷)
1918exp42 424 . . . . . . . . . . 11 (𝜑 → (𝑥𝐴 → (𝑦𝐵 → (𝑦 = 𝐶𝑥 = 𝐷))))
2019com34 91 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 → (𝑦 = 𝐶 → (𝑦𝐵𝑥 = 𝐷))))
2120imp32 407 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
2216, 21jcai 508 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦 = 𝐶)) → (𝑦𝐵𝑥 = 𝐷))
23 eleq1a 2887 . . . . . . . . . . 11 (𝐷𝐴 → (𝑥 = 𝐷𝑥𝐴))
249, 23syl 17 . . . . . . . . . 10 ((𝜑𝑦𝐵) → (𝑥 = 𝐷𝑥𝐴))
2524impr 444 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → 𝑥𝐴)
2617biimpa 464 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑥 = 𝐷) → 𝑦 = 𝐶)
2726exp42 424 . . . . . . . . . . . 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 407 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
3125, 30jcai 508 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥 = 𝐷)) → (𝑥𝐴𝑦 = 𝐶))
3222, 31impbida 826 . . . . . . 7 (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))
3332opabbidv 4917 . . . . . 6 (𝜑 → {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
34 df-mpt 4931 . . . . . . . . 9 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
356, 34syl6eq 2863 . . . . . . . 8 (𝜑𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
3635cnveqd 5506 . . . . . . 7 (𝜑𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
37 cnvopab 5751 . . . . . . 7 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
3836, 37syl6eq 2863 . . . . . 6 (𝜑𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
39 df-mpt 4931 . . . . . . 7 (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)}
4039a1i 11 . . . . . 6 (𝜑 → (𝑦𝐵𝐷) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = 𝐷)})
4133, 38, 403eqtr4d 2857 . . . . 5 (𝜑𝐹 = (𝑦𝐵𝐷))
4241fneq1d 6195 . . . 4 (𝜑 → (𝐹 Fn 𝐵 ↔ (𝑦𝐵𝐷) Fn 𝐵))
4313, 42mpbird 248 . . 3 (𝜑𝐹 Fn 𝐵)
44 dff1o4 6364 . . 3 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
458, 43, 44sylanbrc 574 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
4645, 41jca 503 1 (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3103  {copab 4913  cmpt 4930  ccnv 5317   Fn wfn 6099  1-1-ontowf1o 6103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111
This theorem is referenced by:  fmptco1f1o  29767  ballotlemsf1o  30906
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