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Theorem f1oresf1o2 42142
 Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
f1oresf1o2.1 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresf1o2.2 (𝜑𝐷𝐴)
f1oresf1o2.3 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
Assertion
Ref Expression
f1oresf1o2 (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 f1oresf1o2
StepHypRef Expression
1 f1oresf1o2.1 . . . 4 (𝜑𝐹:𝐴1-1-onto𝐵)
2 f1of1 6355 . . . 4 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)
31, 2syl 17 . . 3 (𝜑𝐹:𝐴1-1𝐵)
4 f1oresf1o2.2 . . 3 (𝜑𝐷𝐴)
5 f1ores 6370 . . 3 ((𝐹:𝐴1-1𝐵𝐷𝐴) → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
63, 4, 5syl2anc 580 . 2 (𝜑 → (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷))
7 f1ofun 6358 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
81, 7syl 17 . . . . 5 (𝜑 → Fun 𝐹)
9 f1odm 6360 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)
101, 9syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
114, 10sseqtr4d 3838 . . . . 5 (𝜑𝐷 ⊆ dom 𝐹)
12 dfimafn 6470 . . . . 5 ((Fun 𝐹𝐷 ⊆ dom 𝐹) → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
138, 11, 12syl2anc 580 . . . 4 (𝜑 → (𝐹𝐷) = {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦})
14 f1of 6356 . . . . . . . . . . . . . . 15 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
151, 14syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹:𝐴𝐵)
1615adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → 𝐹:𝐴𝐵)
174sselda 3798 . . . . . . . . . . . . 13 ((𝜑𝑥𝐷) → 𝑥𝐴)
1816, 17jca 508 . . . . . . . . . . . 12 ((𝜑𝑥𝐷) → (𝐹:𝐴𝐵𝑥𝐴))
19183adant3 1163 . . . . . . . . . . 11 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹:𝐴𝐵𝑥𝐴))
20 ffvelrn 6583 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
2119, 20syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
22 eleq1 2866 . . . . . . . . . . 11 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
23223ad2ant3 1166 . . . . . . . . . 10 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
2421, 23mpbid 224 . . . . . . . . 9 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
25 eqcom 2806 . . . . . . . . . . . 12 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
26 f1oresf1o2.3 . . . . . . . . . . . . . 14 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
2726biimpd 221 . . . . . . . . . . . . 13 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
2827ex 402 . . . . . . . . . . . 12 (𝜑 → (𝑦 = (𝐹𝑥) → (𝑥𝐷𝜒)))
2925, 28syl5bi 234 . . . . . . . . . . 11 (𝜑 → ((𝐹𝑥) = 𝑦 → (𝑥𝐷𝜒)))
3029com23 86 . . . . . . . . . 10 (𝜑 → (𝑥𝐷 → ((𝐹𝑥) = 𝑦𝜒)))
31303imp 1138 . . . . . . . . 9 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝜒)
3224, 31jca 508 . . . . . . . 8 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝑦𝐵𝜒))
3332rexlimdv3a 3214 . . . . . . 7 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 → (𝑦𝐵𝜒)))
34 f1ofo 6363 . . . . . . . . . . . 12 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
351, 34syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐴onto𝐵)
36 foelrni 6469 . . . . . . . . . . 11 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
3735, 36sylan 576 . . . . . . . . . 10 ((𝜑𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
3837ex 402 . . . . . . . . 9 (𝜑 → (𝑦𝐵 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
39 nfv 2010 . . . . . . . . . 10 𝑥𝜑
40 nfv 2010 . . . . . . . . . . 11 𝑥𝜒
41 nfre1 3185 . . . . . . . . . . 11 𝑥𝑥𝐷 (𝐹𝑥) = 𝑦
4240, 41nfim 1996 . . . . . . . . . 10 𝑥(𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
43 rspe 3183 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
4443expcom 403 . . . . . . . . . . . . . . . . 17 ((𝐹𝑥) = 𝑦 → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4544eqcoms 2807 . . . . . . . . . . . . . . . 16 (𝑦 = (𝐹𝑥) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4645adantl 474 . . . . . . . . . . . . . . 15 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4726, 46sylbird 252 . . . . . . . . . . . . . 14 ((𝜑𝑦 = (𝐹𝑥)) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4847ex 402 . . . . . . . . . . . . 13 (𝜑 → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4948adantr 473 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
5025, 49syl5bi 234 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
5150ex 402 . . . . . . . . . 10 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))))
5239, 42, 51rexlimd 3207 . . . . . . . . 9 (𝜑 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
5338, 52syld 47 . . . . . . . 8 (𝜑 → (𝑦𝐵 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
5453impd 399 . . . . . . 7 (𝜑 → ((𝑦𝐵𝜒) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
5533, 54impbid 204 . . . . . 6 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
5655abbidv 2918 . . . . 5 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦 ∣ (𝑦𝐵𝜒)})
57 df-rab 3098 . . . . 5 {𝑦𝐵𝜒} = {𝑦 ∣ (𝑦𝐵𝜒)}
5856, 57syl6eqr 2851 . . . 4 (𝜑 → {𝑦 ∣ ∃𝑥𝐷 (𝐹𝑥) = 𝑦} = {𝑦𝐵𝜒})
5913, 58eqtr2d 2834 . . 3 (𝜑 → {𝑦𝐵𝜒} = (𝐹𝐷))
6059f1oeq3d 6353 . 2 (𝜑 → ((𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹𝐷):𝐷1-1-onto→(𝐹𝐷)))
616, 60mpbird 249 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  {cab 2785  ∃wrex 3090  {crab 3093   ⊆ wss 3769  dom cdm 5312   ↾ cres 5314   “ cima 5315  Fun wfun 6095  ⟶wf 6097  –1-1→wf1 6098  –onto→wfo 6099  –1-1-onto→wf1o 6100  ‘cfv 6101 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 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097 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 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109 This theorem is referenced by: (None)
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