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Theorem f1ossf1o 6867
Description: Restricting a bijection, which is a mapping from a restricted class abstraction, to a subset is a bijection. (Contributed by AV, 7-Aug-2022.)
Hypotheses
Ref Expression
f1ossf1o.x 𝑋 = {𝑤𝐴 ∣ (𝜓𝜒)}
f1ossf1o.y 𝑌 = {𝑤𝐴𝜓}
f1ossf1o.f 𝐹 = (𝑥𝑋𝐵)
f1ossf1o.g 𝐺 = (𝑥𝑌𝐵)
f1ossf1o.b (𝜑𝐺:𝑌1-1-onto𝐶)
f1ossf1o.s ((𝜑𝑥𝑌𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))
Assertion
Ref Expression
f1ossf1o (𝜑𝐹:𝑋1-1-onto→{𝑦𝐶𝜏})
Distinct variable groups:   𝑤,𝐴,𝑥   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝑋   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦   𝜓,𝑥   𝜒,𝑥,𝑦   𝜏,𝑥   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑦,𝑤)   𝜒(𝑤)   𝜏(𝑦,𝑤)   𝐴(𝑦)   𝐵(𝑥,𝑤)   𝐶(𝑤)   𝐹(𝑥,𝑦,𝑤)   𝐺(𝑥,𝑦,𝑤)   𝑋(𝑦,𝑤)   𝑌(𝑤)

Proof of Theorem f1ossf1o
StepHypRef Expression
1 f1ossf1o.g . . 3 𝐺 = (𝑥𝑌𝐵)
2 f1ossf1o.b . . 3 (𝜑𝐺:𝑌1-1-onto𝐶)
3 f1ossf1o.s . . 3 ((𝜑𝑥𝑌𝑦 = 𝐵) → (𝜏 ↔ [𝑥 / 𝑤]𝜒))
41, 2, 3f1oresrab 6866 . 2 (𝜑 → (𝐺 ↾ {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}–1-1-onto→{𝑦𝐶𝜏})
5 simpl 486 . . . . . . . . 9 ((𝜓𝜒) → 𝜓)
65a1i 11 . . . . . . . 8 (𝑤𝐴 → ((𝜓𝜒) → 𝜓))
76ss2rabi 4004 . . . . . . 7 {𝑤𝐴 ∣ (𝜓𝜒)} ⊆ {𝑤𝐴𝜓}
8 f1ossf1o.x . . . . . . 7 𝑋 = {𝑤𝐴 ∣ (𝜓𝜒)}
9 f1ossf1o.y . . . . . . 7 𝑌 = {𝑤𝐴𝜓}
107, 8, 93sstr4i 3958 . . . . . 6 𝑋𝑌
1110a1i 11 . . . . 5 (𝜑𝑋𝑌)
1211resmptd 5875 . . . 4 (𝜑 → ((𝑥𝑌𝐵) ↾ 𝑋) = (𝑥𝑋𝐵))
131a1i 11 . . . . 5 (𝜑𝐺 = (𝑥𝑌𝐵))
149rabeqi 3429 . . . . . . 7 {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒} = {𝑥 ∈ {𝑤𝐴𝜓} ∣ [𝑥 / 𝑤]𝜒}
15 nfcv 2955 . . . . . . . . . . 11 𝑤𝑥
16 nfcv 2955 . . . . . . . . . . 11 𝑤𝐴
17 nfs1v 2157 . . . . . . . . . . 11 𝑤[𝑥 / 𝑤]𝜓
18 sbequ12 2250 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑤]𝜓))
1915, 16, 17, 18elrabf 3624 . . . . . . . . . 10 (𝑥 ∈ {𝑤𝐴𝜓} ↔ (𝑥𝐴 ∧ [𝑥 / 𝑤]𝜓))
2019anbi1i 626 . . . . . . . . 9 ((𝑥 ∈ {𝑤𝐴𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ ((𝑥𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒))
21 anass 472 . . . . . . . . 9 (((𝑥𝐴 ∧ [𝑥 / 𝑤]𝜓) ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
2220, 21bitri 278 . . . . . . . 8 ((𝑥 ∈ {𝑤𝐴𝜓} ∧ [𝑥 / 𝑤]𝜒) ↔ (𝑥𝐴 ∧ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
2322rabbia2 3424 . . . . . . 7 {𝑥 ∈ {𝑤𝐴𝜓} ∣ [𝑥 / 𝑤]𝜒} = {𝑥𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
24 nfcv 2955 . . . . . . . . 9 𝑥𝐴
25 nfv 1915 . . . . . . . . 9 𝑥(𝜓𝜒)
26 nfs1v 2157 . . . . . . . . . 10 𝑤[𝑥 / 𝑤]𝜒
2717, 26nfan 1900 . . . . . . . . 9 𝑤([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)
28 sbequ12 2250 . . . . . . . . . 10 (𝑤 = 𝑥 → (𝜒 ↔ [𝑥 / 𝑤]𝜒))
2918, 28anbi12d 633 . . . . . . . . 9 (𝑤 = 𝑥 → ((𝜓𝜒) ↔ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)))
3016, 24, 25, 27, 29cbvrabw 3437 . . . . . . . 8 {𝑤𝐴 ∣ (𝜓𝜒)} = {𝑥𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)}
318, 30eqtr2i 2822 . . . . . . 7 {𝑥𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = 𝑋
3214, 23, 313eqtri 2825 . . . . . 6 {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋
3332a1i 11 . . . . 5 (𝜑 → {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒} = 𝑋)
3413, 33reseq12d 5819 . . . 4 (𝜑 → (𝐺 ↾ {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}) = ((𝑥𝑌𝐵) ↾ 𝑋))
35 f1ossf1o.f . . . . 5 𝐹 = (𝑥𝑋𝐵)
3635a1i 11 . . . 4 (𝜑𝐹 = (𝑥𝑋𝐵))
3712, 34, 363eqtr4rd 2844 . . 3 (𝜑𝐹 = (𝐺 ↾ {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}))
3814, 23eqtr2i 2822 . . . . 5 {𝑥𝐴 ∣ ([𝑥 / 𝑤]𝜓 ∧ [𝑥 / 𝑤]𝜒)} = {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}
398, 30, 383eqtri 2825 . . . 4 𝑋 = {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}
4039a1i 11 . . 3 (𝜑𝑋 = {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒})
41 eqidd 2799 . . 3 (𝜑 → {𝑦𝐶𝜏} = {𝑦𝐶𝜏})
4237, 40, 41f1oeq123d 6585 . 2 (𝜑 → (𝐹:𝑋1-1-onto→{𝑦𝐶𝜏} ↔ (𝐺 ↾ {𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}):{𝑥𝑌 ∣ [𝑥 / 𝑤]𝜒}–1-1-onto→{𝑦𝐶𝜏}))
434, 42mpbird 260 1 (𝜑𝐹:𝑋1-1-onto→{𝑦𝐶𝜏})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  [wsb 2069  wcel 2111  {crab 3110  wss 3881  cmpt 5110  cres 5521  1-1-ontowf1o 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332
This theorem is referenced by:  clwwlknonclwlknonf1o  28147  dlwwlknondlwlknonf1o  28150
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