Theorem bj-gabima 37426

Description: Generalized class abstraction as a direct image.

TODO: improve the support lemmas elimag 6054 and fvelima 6933 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 2926	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝑦)
3		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
5		df-rex 3088	. . . . . 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 2770	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑧) ↔ (𝐹‘𝑧) = 𝑦)
8		df-clab 2742	. . . . . . . . 9 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜓} ↔ [𝑧 / 𝑥]𝜓)
9	8	bicomi 226	. . . . . . . 8 ⊢ ([𝑧 / 𝑥]𝜓 ↔ 𝑧 ∈ {𝑥 ∣ 𝜓})
10	7, 9	anbi12ci 638	. . . . . . 7 ⊢ ((𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ (𝑧 ∈ {𝑥 ∣ 𝜓} ∧ (𝐹‘𝑧) = 𝑦))
11	10	exbii 1869	. . . . . 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 2181	. . . . . 6 ⊢ Ⅎ𝑥𝜑
14		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥𝑦)
16		bj-gabima.nff	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝐹)
17		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → Ⅎ𝑥𝑧)
19	16, 18	nffvd 6880	. . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝑧))
20	15, 19	nfeqd 2935	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = (𝐹‘𝑧))
21		nfs1v 2191	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑥]𝜓)
23	20, 22	nfand 1918	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓))
24		fveq2 6868	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
25	24	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 = (𝐹‘𝑧) ↔ 𝑦 = (𝐹‘𝑥)))
26		sbequ12r 2288	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥]𝜓 ↔ 𝜓))
27	25, 26	anbi12d 641	. . . . . . 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 2371	. . . . 5 ⊢ (𝜑 → (∃𝑧(𝑦 = (𝐹‘𝑧) ∧ [𝑧 / 𝑥]𝜓) ↔ ∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓)))
30	6, 12, 29	3bitr2rd 310	. . . 4 ⊢ (𝜑 → (∃𝑥(𝑦 = (𝐹‘𝑥) ∧ 𝜓) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
31	1, 2, 4, 30	bj-elgab 37425	. . 3 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
32		bj-gabima.fun	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
33	32	funfnd 6553	. . . 4 ⊢ (𝜑 → 𝐹 Fn dom 𝐹)
34		bj-gabima.dm	. . . 4 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ dom 𝐹)
35	33, 34	fvelimabd 6941	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓}) ↔ ∃𝑧 ∈ {𝑥 ∣ 𝜓} (𝐹‘𝑧) = 𝑦))
36	31, 35	bitr4d 284	. 2 ⊢ (𝜑 → (𝑦 ∈ {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} ↔ 𝑦 ∈ (𝐹 “ {𝑥 ∣ 𝜓})))
37	36	eqrdv 2761	1 ⊢ (𝜑 → {(𝐹‘𝑥) ∣ 𝑥 ∣ 𝜓} = (𝐹 “ {𝑥 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1559   = wceq 1561  ∃wex 1800  Ⅎwnf 1804  [wsb 2091   ∈ wcel 2143  {cab 2741  Ⅎwnfc 2910  ∃wrex 3087  Vcvv 3455   ⊆ wss 3905  dom cdm 5648   “ cima 5651  Fun wfun 6516  ‘cfv 6522  {bj-cgab 37419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-fv 6530  df-bj-gab 37420

This theorem is referenced by: (None)