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Theorem bj-gabima 36923
Description: Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 6084 and fvelima 6974 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 3482 . . . . 5 𝑦 ∈ V
43a1i 11 . . . 4 (𝜑𝑦 ∈ V)
5 df-rex 3069 . . . . . 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 2742 . . . . . . . 8 (𝑦 = (𝐹𝑧) ↔ (𝐹𝑧) = 𝑦)
8 df-clab 2713 . . . . . . . . 9 (𝑧 ∈ {𝑥𝜓} ↔ [𝑧 / 𝑥]𝜓)
98bicomi 224 . . . . . . . 8 ([𝑧 / 𝑥]𝜓𝑧 ∈ {𝑥𝜓})
107, 9anbi12ci 629 . . . . . . 7 ((𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑧 ∈ {𝑥𝜓} ∧ (𝐹𝑧) = 𝑦))
1110exbii 1845 . . . . . 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 2144 . . . . . 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 6919 . . . . . . . 8 (𝜑𝑥(𝐹𝑧))
2015, 19nfeqd 2914 . . . . . . 7 (𝜑 → Ⅎ𝑥 𝑦 = (𝐹𝑧))
21 nfs1v 2154 . . . . . . . 8 𝑥[𝑧 / 𝑥]𝜓
2221a1i 11 . . . . . . 7 (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
2320, 22nfand 1895 . . . . . 6 (𝜑 → Ⅎ𝑥(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓))
24 fveq2 6907 . . . . . . . . 9 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2524eqeq2d 2746 . . . . . . . 8 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
26 sbequ12r 2250 . . . . . . . 8 (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜓𝜓))
2725, 26anbi12d 632 . . . . . . 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 2340 . . . . 5 (𝜑 → (∃𝑧(𝑦 = (𝐹𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑥(𝑦 = (𝐹𝑥) ∧ 𝜓)))
306, 12, 293bitr2rd 308 . . . 4 (𝜑 → (∃𝑥(𝑦 = (𝐹𝑥) ∧ 𝜓) ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
311, 2, 4, 30bj-elgab 36922 . . 3 (𝜑 → (𝑦 ∈ {(𝐹𝑥) ∣ 𝑥𝜓} ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
32 bj-gabima.fun . . . . 5 (𝜑 → Fun 𝐹)
3332funfnd 6599 . . . 4 (𝜑𝐹 Fn dom 𝐹)
34 bj-gabima.dm . . . 4 (𝜑 → {𝑥𝜓} ⊆ dom 𝐹)
3533, 34fvelimabd 6982 . . 3 (𝜑 → (𝑦 ∈ (𝐹 “ {𝑥𝜓}) ↔ ∃𝑧 ∈ {𝑥𝜓} (𝐹𝑧) = 𝑦))
3631, 35bitr4d 282 . 2 (𝜑 → (𝑦 ∈ {(𝐹𝑥) ∣ 𝑥𝜓} ↔ 𝑦 ∈ (𝐹 “ {𝑥𝜓})))
3736eqrdv 2733 1 (𝜑 → {(𝐹𝑥) ∣ 𝑥𝜓} = (𝐹 “ {𝑥𝜓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wnf 1780  [wsb 2062  wcel 2106  {cab 2712  wnfc 2888  wrex 3068  Vcvv 3478  wss 3963  dom cdm 5689  cima 5692  Fun wfun 6557  cfv 6563  {bj-cgab 36916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-bj-gab 36917
This theorem is referenced by: (None)
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