Theorem fvmptex 7043

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 6933.) 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 3924	. . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmptex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3935	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5277	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2768	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmpti 7028	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
9	1	fveq2d 6924	. . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
10		fvmptex.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))
11		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵)
12		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥 I
13	12, 4	nffv 6930	. . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵)
14	5	fveq2d 6924	. . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵))
15	11, 13, 14	cbvmpt 5277	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
16	10, 15	eqtri 2768	. . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
17		fvex 6933	. . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V
18	9, 16, 17	fvmpt 7029	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
19	8, 18	eqtr4d 2783	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
20	2	dmmptss 6272	. . . . 5 ⊢ dom 𝐹 ⊆ 𝐴
21	20	sseli 4004	. . . 4 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴)
22		ndmfv 6955	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅)
23	21, 22	nsyl5 159	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅)
24		fvex 6933	. . . . . 6 ⊢ ( I ‘𝐵) ∈ V
25	24, 10	dmmpti 6724	. . . . 5 ⊢ dom 𝐺 = 𝐴
26	25	eleq2i 2836	. . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴)
27		ndmfv 6955	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅)
28	26, 27	sylnbir 331	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅)
29	23, 28	eqtr4d 2783	. 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
30	19, 29	pm2.61i 182	1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  ⦋csb 3921  ∅c0 4352   ↦ cmpt 5249   I cid 5592  dom cdm 5700  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  fvmptnf  7051  sumeq2ii  15741  prodeq2ii  15959