Theorem fvmptex 5721

Description: Express a function F whose value B may not always be a set in terms of another function G for which sethood is guaranteed. (Note that ( I ‘B) is just shorthand for if(B ∈ V, B, ∅), and it is always a set by fvex 5339.) Note also that these functions are not the same; wherever B(C) is not a set, C is not in the domain of F (so it evaluates to the empty set), but C is in the domain of G, and G(C) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fvmptex.1	⊢ F = (x ∈ A ↦ B)
fvmptex.2	⊢ G = (x ∈ A ↦ ( I ‘B))

Assertion

Ref	Expression
fvmptex	⊢ (F ‘C) = (G ‘C)

Distinct variable group:   x,A

Allowed substitution hints:   B(x)   C(x)   F(x)   G(x)

Proof of Theorem fvmptex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3139	. . . 4 ⊢ (y = C → [y / x]B = [C / x]B)
2		fvmptex.1	. . . . 5 ⊢ F = (x ∈ A ↦ B)
3		nfcv 2489	. . . . . 6 ⊢ ℲyB
4		nfcsb1v 3168	. . . . . 6 ⊢ Ⅎx[y / x]B
5		csbeq1a 3144	. . . . . 6 ⊢ (x = y → B = [y / x]B)
6	3, 4, 5	cbvmpt 5676	. . . . 5 ⊢ (x ∈ A ↦ B) = (y ∈ A ↦ [y / x]B)
7	2, 6	eqtri 2373	. . . 4 ⊢ F = (y ∈ A ↦ [y / x]B)
8	1, 7	fvmpti 5699	. . 3 ⊢ (C ∈ A → (F ‘C) = ( I ‘[C / x]B))
9	1	fveq2d 5332	. . . 4 ⊢ (y = C → ( I ‘[y / x]B) = ( I ‘[C / x]B))
10		fvmptex.2	. . . . 5 ⊢ G = (x ∈ A ↦ ( I ‘B))
11		nfcv 2489	. . . . . 6 ⊢ Ⅎy( I ‘B)
12		nfcv 2489	. . . . . . 7 ⊢ Ⅎx I
13	12, 4	nffv 5334	. . . . . 6 ⊢ Ⅎx( I ‘[y / x]B)
14	5	fveq2d 5332	. . . . . 6 ⊢ (x = y → ( I ‘B) = ( I ‘[y / x]B))
15	11, 13, 14	cbvmpt 5676	. . . . 5 ⊢ (x ∈ A ↦ ( I ‘B)) = (y ∈ A ↦ ( I ‘[y / x]B))
16	10, 15	eqtri 2373	. . . 4 ⊢ G = (y ∈ A ↦ ( I ‘[y / x]B))
17		fvex 5339	. . . 4 ⊢ ( I ‘[C / x]B) ∈ V
18	9, 16, 17	fvmpt 5700	. . 3 ⊢ (C ∈ A → (G ‘C) = ( I ‘[C / x]B))
19	8, 18	eqtr4d 2388	. 2 ⊢ (C ∈ A → (F ‘C) = (G ‘C))
20	2	dmmptss 5685	. . . . . 6 ⊢ dom F ⊆ A
21	20	sseli 3269	. . . . 5 ⊢ (C ∈ dom F → C ∈ A)
22	21	con3i 127	. . . 4 ⊢ (¬ C ∈ A → ¬ C ∈ dom F)
23		ndmfv 5349	. . . 4 ⊢ (¬ C ∈ dom F → (F ‘C) = ∅)
24	22, 23	syl 15	. . 3 ⊢ (¬ C ∈ A → (F ‘C) = ∅)
25		fvex 5339	. . . . . 6 ⊢ ( I ‘B) ∈ V
26	25, 10	dmmpti 5691	. . . . 5 ⊢ dom G = A
27	26	eleq2i 2417	. . . 4 ⊢ (C ∈ dom G ↔ C ∈ A)
28		ndmfv 5349	. . . 4 ⊢ (¬ C ∈ dom G → (G ‘C) = ∅)
29	27, 28	sylnbir 298	. . 3 ⊢ (¬ C ∈ A → (G ‘C) = ∅)
30	24, 29	eqtr4d 2388	. 2 ⊢ (¬ C ∈ A → (F ‘C) = (G ‘C))
31	19, 30	pm2.61i 156	1 ⊢ (F ‘C) = (G ‘C)

Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  [csb 3136  ∅c0 3550   I cid 4763  dom cdm 4772   ‘cfv 4781   ↦ cmpt 5651

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795  df-mpt 5652

This theorem is referenced by:  fvmptnf  5723