Theorem fvmptex 6785

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 6686.) 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 3889	. . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmptex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2980	. . . . . 6 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3910	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3900	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5170	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2847	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmpti 6770	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
9	1	fveq2d 6677	. . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
10		fvmptex.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))
11		nfcv 2980	. . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵)
12		nfcv 2980	. . . . . . 7 ⊢ Ⅎ𝑥 I
13	12, 4	nffv 6683	. . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵)
14	5	fveq2d 6677	. . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵))
15	11, 13, 14	cbvmpt 5170	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
16	10, 15	eqtri 2847	. . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
17		fvex 6686	. . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V
18	9, 16, 17	fvmpt 6771	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
19	8, 18	eqtr4d 2862	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
20	2	dmmptss 6098	. . . . . 6 ⊢ dom 𝐹 ⊆ 𝐴
21	20	sseli 3966	. . . . 5 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴)
22	21	con3i 157	. . . 4 ⊢ (¬ 𝐶 ∈ 𝐴 → ¬ 𝐶 ∈ dom 𝐹)
23		ndmfv 6703	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅)
24	22, 23	syl 17	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅)
25		fvex 6686	. . . . . 6 ⊢ ( I ‘𝐵) ∈ V
26	25, 10	dmmpti 6495	. . . . 5 ⊢ dom 𝐺 = 𝐴
27	26	eleq2i 2907	. . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴)
28		ndmfv 6703	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅)
29	27, 28	sylnbir 333	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅)
30	24, 29	eqtr4d 2862	. 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
31	19, 30	pm2.61i 184	1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  ⦋csb 3886  ∅c0 4294   ↦ cmpt 5149   I cid 5462  dom cdm 5558  ‘cfv 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366

This theorem is referenced by:  fvmptnf  6793  sumeq2ii  15053  prodeq2ii  15270