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Theorem bj-gabima 35906
Description: Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 6063 and fvelima 6957 to nonfreeness hypothesis (and for the latter, biconditional). (Contributed by BJ, 4-Oct-2024.)

Hypotheses
Ref Expression
bj-gabima.nf (𝜑 → ∀𝑥𝜑)
bj-gabima.nff (𝜑𝑥𝐹)
bj-gabima.fun (𝜑 → Fun 𝐹)
bj-gabima.dm (𝜑 → {𝑥𝜓} ⊆ dom 𝐹)
Assertion
Ref Expression
bj-gabima (𝜑 → {(𝐹𝑥) ∣ 𝑥𝜓} = (𝐹 “ {𝑥𝜓}))

Proof of Theorem bj-gabima
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-gabima.nf . . . 4 (𝜑 → ∀𝑥𝜑)
2 nfcvd 2904 . . . 4 (𝜑𝑥𝑦)
3 vex 3478 . . . . 5 𝑦 ∈ V
43a1i 11 . . . 4 (𝜑𝑦 ∈ V)
5 df-rex 3071 . . . . . 6 (∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦 ↔ ∃𝑧(𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦))
65a1i 11 . . . . 5 (𝜑 → (∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦 ↔ ∃𝑧(𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦)))
7 eqcom 2739 . . . . . . . 8 (𝑦 = (𝐹𝑧) ↔ (𝐹𝑧) = 𝑦)
8 df-clab 2710 . . . . . . . . 9 (𝑧 ∈ {𝑥𝜓} ↔ [𝑧 / 𝑥]𝜓)
98bicomi 223 . . . . . . . 8 ([𝑧 / 𝑥]𝜓𝑧 ∈ {𝑥𝜓})
107, 9anbi12ci 628 . . . . . . 7 ((𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦))
1110exbii 1850 . . . . . 6 (∃𝑧(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑧(𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦))
1211a1i 11 . . . . 5 (𝜑 → (∃𝑧(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑧(𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦)))
131nf5i 2142 . . . . . 6 𝑥𝜑
14 nfcv 2903 . . . . . . . . 9 𝑥𝑦
1514a1i 11 . . . . . . . 8 (𝜑𝑥𝑦)
16 bj-gabima.nff . . . . . . . . 9 (𝜑𝑥𝐹)
17 nfcv 2903 . . . . . . . . . 10 𝑥𝑧
1817a1i 11 . . . . . . . . 9 (𝜑𝑥𝑧)
1916, 18nffvd 6903 . . . . . . . 8 (𝜑𝑥(𝐹𝑧))
2015, 19nfeqd 2913 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = (𝐹𝑧))
21 nfs1v 2153 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜓
2221a1i 11 . . . . . . 7 (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
2320, 22nfand 1900 . . . . . 6 (𝜑 → Ⅎ𝑥(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓))
24 fveq2 6891 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2524eqeq2d 2743 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
26 sbequ12r 2244 . . . . . . . 8 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜓𝜓))
2725, 26anbi12d 631 . . . . . . 7 (𝑧 = 𝑥 → ((𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑦 = (𝐹𝑥) ∧ 𝜓)))
2827a1i 11 . . . . . 6 (𝜑 → (𝑧 = 𝑥 → ((𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑦 = (𝐹𝑥) ∧ 𝜓))))
2913, 23, 28cbvexdw 2335 . . . . 5 (𝜑 → (∃𝑧(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑥(𝑦 = (𝐹𝑥) ∧ 𝜓)))
306, 12, 293bitr2rd 307 . . . 4 (𝜑 → (∃𝑥(𝑦 = (𝐹𝑥) ∧ 𝜓) ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
311, 2, 4, 30bj-elgab 35905 . . 3 (𝜑 → (𝑦 ∈ {(𝐹𝑥) ∣ 𝑥𝜓} ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
32 bj-gabima.fun . . . . 5 (𝜑 → Fun 𝐹)
3332funfnd 6579 . . . 4 (𝜑𝐹 Fn dom 𝐹)
34 bj-gabima.dm . . . 4 (𝜑 → {𝑥𝜓} ⊆ dom 𝐹)
3533, 34fvelimabd 6965 . . 3 (𝜑 → (𝑦 ∈ (𝐹 “ {𝑥𝜓}) ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
3631, 35bitr4d 281 . 2 (𝜑 → (𝑦 ∈ {(𝐹𝑥) ∣ 𝑥𝜓} ↔ 𝑦 ∈ (𝐹 “ {𝑥𝜓})))
3736eqrdv 2730 1 (𝜑 → {(𝐹𝑥) ∣ 𝑥𝜓} = (𝐹 “ {𝑥𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wex 1781  wnf 1785  [wsb 2067  wcel 2106  {cab 2709  wnfc 2883  wrex 3070  Vcvv 3474  wss 3948  dom cdm 5676  cima 5679  Fun wfun 6537  cfv 6543  {bj-cgab 35899
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-bj-gab 35900
This theorem is referenced by: (None)
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