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Theorem f1oresf1o 42140
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
f1oresf1o.1 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresf1o.2 (𝜑𝐷𝐴)
f1oresf1o.3 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
Assertion
Ref Expression
f1oresf1o (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem f1oresf1o
StepHypRef Expression
1 f1oresf1o.1 . . . 4 (𝜑𝐹:𝐴1-1-onto𝐵)
2 f1of1 6356 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
31, 2syl 17 . . 3 (𝜑𝐹:𝐴1-1𝐵)
4 f1oresf1o.2 . . 3 (𝜑𝐷𝐴)
5 f1ores 6371 . . 3 ((𝐹:𝐴1-1𝐵𝐷𝐴) → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
63, 4, 5syl2anc 580 . 2 (𝜑 → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
7 f1ofun 6359 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
81, 7syl 17 . . . . 5 (𝜑 → Fun 𝐹)
9 f1odm 6361 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
101, 9syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
114, 10sseqtr4d 3839 . . . . 5 (𝜑𝐷 ⊆ dom 𝐹)
12 dfimafn 6471 . . . . 5 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
138, 11, 12syl2anc 580 . . . 4 (𝜑 → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
14 f1oresf1o.3 . . . . . 6 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
1514abbidv 2919 . . . . 5 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦 ∣ (𝑦𝐵𝜒)})
16 df-rab 3099 . . . . 5 {𝑦𝐵𝜒} = {𝑦 ∣ (𝑦𝐵𝜒)}
1715, 16syl6eqr 2852 . . . 4 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦𝐵𝜒})
1813, 17eqtr2d 2835 . . 3 (𝜑 → {𝑦𝐵𝜒} = (𝐹𝐷))
1918f1oeq3d 6354 . 2 (𝜑 → ((𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷)))
206, 19mpbird 249 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {cab 2786  wrex 3091  {crab 3094  wss 3770  dom cdm 5313  cres 5315  cima 5316  Fun wfun 6096  1-1wf1 6099  1-1-ontowf1o 6101  cfv 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110
This theorem is referenced by: (None)
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