Theorem cbvmpt2x 5679

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5680 allows B to be a function of x. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
cbvmpt2x.1	⊢ ℲzB
cbvmpt2x.2	⊢ ℲxD
cbvmpt2x.3	⊢ ℲzC
cbvmpt2x.4	⊢ ℲwC
cbvmpt2x.5	⊢ ℲxE
cbvmpt2x.6	⊢ ℲyE
cbvmpt2x.7	⊢ (x = z → B = D)
cbvmpt2x.8	⊢ ((x = z ∧ y = w) → C = E)

Assertion

Ref	Expression
cbvmpt2x	⊢ (x ∈ A, y ∈ B ↦ C) = (z ∈ A, w ∈ D ↦ E)

Distinct variable groups:   x,w,y,z,A   w,B   y,D

Allowed substitution hints:   B(x,y,z)   C(x,y,z,w)   D(x,z,w)   E(x,y,z,w)

Proof of Theorem cbvmpt2x

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . 5 ⊢ Ⅎz x ∈ A
2		cbvmpt2x.1	. . . . . 6 ⊢ ℲzB
3	2	nfcri 2484	. . . . 5 ⊢ Ⅎz y ∈ B
4	1, 3	nfan 1824	. . . 4 ⊢ Ⅎz(x ∈ A ∧ y ∈ B)
5		cbvmpt2x.3	. . . . 5 ⊢ ℲzC
6	5	nfeq2 2501	. . . 4 ⊢ Ⅎz u = C
7	4, 6	nfan 1824	. . 3 ⊢ Ⅎz((x ∈ A ∧ y ∈ B) ∧ u = C)
8		nfv 1619	. . . . 5 ⊢ Ⅎw x ∈ A
9		nfcv 2490	. . . . . 6 ⊢ ℲwB
10	9	nfcri 2484	. . . . 5 ⊢ Ⅎw y ∈ B
11	8, 10	nfan 1824	. . . 4 ⊢ Ⅎw(x ∈ A ∧ y ∈ B)
12		cbvmpt2x.4	. . . . 5 ⊢ ℲwC
13	12	nfeq2 2501	. . . 4 ⊢ Ⅎw u = C
14	11, 13	nfan 1824	. . 3 ⊢ Ⅎw((x ∈ A ∧ y ∈ B) ∧ u = C)
15		nfv 1619	. . . . 5 ⊢ Ⅎx z ∈ A
16		cbvmpt2x.2	. . . . . 6 ⊢ ℲxD
17	16	nfcri 2484	. . . . 5 ⊢ Ⅎx w ∈ D
18	15, 17	nfan 1824	. . . 4 ⊢ Ⅎx(z ∈ A ∧ w ∈ D)
19		cbvmpt2x.5	. . . . 5 ⊢ ℲxE
20	19	nfeq2 2501	. . . 4 ⊢ Ⅎx u = E
21	18, 20	nfan 1824	. . 3 ⊢ Ⅎx((z ∈ A ∧ w ∈ D) ∧ u = E)
22		nfv 1619	. . . 4 ⊢ Ⅎy(z ∈ A ∧ w ∈ D)
23		cbvmpt2x.6	. . . . 5 ⊢ ℲyE
24	23	nfeq2 2501	. . . 4 ⊢ Ⅎy u = E
25	22, 24	nfan 1824	. . 3 ⊢ Ⅎy((z ∈ A ∧ w ∈ D) ∧ u = E)
26		eleq1 2413	. . . . . 6 ⊢ (x = z → (x ∈ A ↔ z ∈ A))
27	26	adantr 451	. . . . 5 ⊢ ((x = z ∧ y = w) → (x ∈ A ↔ z ∈ A))
28		cbvmpt2x.7	. . . . . . 7 ⊢ (x = z → B = D)
29	28	eleq2d 2420	. . . . . 6 ⊢ (x = z → (y ∈ B ↔ y ∈ D))
30		eleq1 2413	. . . . . 6 ⊢ (y = w → (y ∈ D ↔ w ∈ D))
31	29, 30	sylan9bb 680	. . . . 5 ⊢ ((x = z ∧ y = w) → (y ∈ B ↔ w ∈ D))
32	27, 31	anbi12d 691	. . . 4 ⊢ ((x = z ∧ y = w) → ((x ∈ A ∧ y ∈ B) ↔ (z ∈ A ∧ w ∈ D)))
33		cbvmpt2x.8	. . . . 5 ⊢ ((x = z ∧ y = w) → C = E)
34	33	eqeq2d 2364	. . . 4 ⊢ ((x = z ∧ y = w) → (u = C ↔ u = E))
35	32, 34	anbi12d 691	. . 3 ⊢ ((x = z ∧ y = w) → (((x ∈ A ∧ y ∈ B) ∧ u = C) ↔ ((z ∈ A ∧ w ∈ D) ∧ u = E)))
36	7, 14, 21, 25, 35	cbvoprab12 5570	. 2 ⊢ {⟨⟨x, y⟩, u⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ u = C)} = {⟨⟨z, w⟩, u⟩ ∣ ((z ∈ A ∧ w ∈ D) ∧ u = E)}
37		df-mpt2 5655	. 2 ⊢ (x ∈ A, y ∈ B ↦ C) = {⟨⟨x, y⟩, u⟩ ∣ ((x ∈ A ∧ y ∈ B) ∧ u = C)}
38		df-mpt2 5655	. 2 ⊢ (z ∈ A, w ∈ D ↦ E) = {⟨⟨z, w⟩, u⟩ ∣ ((z ∈ A ∧ w ∈ D) ∧ u = E)}
39	36, 37, 38	3eqtr4i 2383	1 ⊢ (x ∈ A, y ∈ B ↦ C) = (z ∈ A, w ∈ D ↦ E)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  {coprab 5528   ↦ cmpt2 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-oprab 5529  df-mpt2 5655

This theorem is referenced by:  cbvmpt2  5680