Theorem bj-gabima 37143

Description: Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 6023 and fvelima 6899 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 2899	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝑦)
3		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
5		df-rex 3061	. . . . . 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 2743	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) ↔ (𝐹‘𝑧) = 𝑦)
8		df-clab 2715	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜓} ↔ [𝑧 / 𝑥]𝜓)
9	8	bicomi 224	. . . . . . . 8 ⊢ ([𝑧 / 𝑥]𝜓 ↔ 𝑧 ∈ {𝑥 ∣ 𝜓})
10	7, 9	anbi12ci 629	. . . . . . 7 ⊢ ((𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦))
11	10	exbii 1849	. . . . . 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 2151	. . . . . 6 ⊢ Ⅎ𝑥𝜑
14		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
16		bj-gabima.nff	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝐹)
17		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝑧)
19	16, 18	nffvd 6846	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝑧))
20	15, 19	nfeqd 2909	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = (𝐹‘𝑧))
21		nfs1v 2161	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
23	20, 22	nfand 1898	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓))
24		fveq2 6834	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
25	24	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥)))
26		sbequ12r 2259	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜓 ↔ 𝜓))
27	25, 26	anbi12d 632	. . . . . . 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 2343	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓)))
30	6, 12, 29	3bitr2rd 308	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
31	1, 2, 4, 30	bj-elgab 37142	. . 3 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
32		bj-gabima.fun	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
33	32	funfnd 6523	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
34		bj-gabima.dm	. . . 4 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)
35	33, 34	fvelimabd 6907	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓}) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
36	31, 35	bitr4d 282	. 2 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ 𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓})))
37	36	eqrdv 2734	1 ⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780  Ⅎwnf 1784  [wsb 2067   ∈ wcel 2113  {cab 2714  Ⅎwnfc 2883  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  dom cdm 5624   “ cima 5627  Fun wfun 6486  ‘cfv 6492  {bj-cgab 37136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500  df-bj-gab 37137

This theorem is referenced by: (None)