Theorem bj-gabima 35107

Description: Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 5970 and fvelima 6829 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	⊢ (𝜑 → Ⅎ𝑥𝐹)
bj-gabima.fun	⊢ (𝜑 → Fun 𝐹)
bj-gabima.dm	⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)

Assertion

Ref	Expression
bj-gabima	⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))

Proof of Theorem bj-gabima

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-gabima.nf	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
2		nfcvd 2909	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝑦)
3		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
5		df-rex 3071	. . . . . 6 ⊢ (∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦 ↔ ∃𝑧(𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦))
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦 ↔ ∃𝑧(𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦)))
7		eqcom 2746	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) ↔ (𝐹‘𝑧) = 𝑦)
8		df-clab 2717	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜓} ↔ [𝑧 / 𝑥]𝜓)
9	8	bicomi 223	. . . . . . . 8 ⊢ ([𝑧 / 𝑥]𝜓 ↔ 𝑧 ∈ {𝑥 ∣ 𝜓})
10	7, 9	anbi12ci 627	. . . . . . 7 ⊢ ((𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦))
11	10	exbii 1853	. . . . . 6 ⊢ (∃𝑧(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑧(𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦))
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑧(𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦)))
13	1	nf5i 2145	. . . . . 6 ⊢ Ⅎ𝑥𝜑
14		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
16		bj-gabima.nff	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝐹)
17		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝑧)
19	16, 18	nffvd 6780	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝑧))
20	15, 19	nfeqd 2918	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = (𝐹‘𝑧))
21		nfs1v 2156	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
23	20, 22	nfand 1903	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓))
24		fveq2 6768	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
25	24	eqeq2d 2750	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥)))
26		sbequ12r 2248	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜓 ↔ 𝜓))
27	25, 26	anbi12d 630	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑦 = (𝐹‘𝑥) ∧ 𝜓)))
28	27	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑧 = 𝑥 → ((𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑦 = (𝐹‘𝑥) ∧ 𝜓))))
29	13, 23, 28	cbvexdw 2339	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓)))
30	6, 12, 29	3bitr2rd 307	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
31	1, 2, 4, 30	bj-elgab 35106	. . 3 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
32		bj-gabima.fun	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
33	32	funfnd 6461	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
34		bj-gabima.dm	. . . 4 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)
35	33, 34	fvelimabd 6836	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓}) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
36	31, 35	bitr4d 281	. 2 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ 𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓})))
37	36	eqrdv 2737	1 ⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1785  Ⅎwnf 1789  [wsb 2070   ∈ wcel 2109  {cab 2716  Ⅎwnfc 2888  ∃wrex 3066  Vcvv 3430   ⊆ wss 3891  dom cdm 5588   “ cima 5591  Fun wfun 6424  ‘cfv 6430  {bj-cgab 35100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-fv 6438  df-bj-gab 35101

This theorem is referenced by: (None)