Theorem fvmpopr2d 7569

Description: Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.)

Hypotheses

Ref	Expression
fvmpopr2d.1	⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
fvmpopr2d.2	⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
fvmpopr2d.3	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fvmpopr2d	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Distinct variable groups:   𝐴,𝑎,𝑏   𝐵,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑃(𝑎,𝑏)   𝐹(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem fvmpopr2d

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7408	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩)
2		fvmpopr2d.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
4		fvmpopr2d.2	. . . . 5 ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
5	4	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
6	3, 5	fveq12d 6883	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩))
7	1, 6	eqtr4id 2789	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝐹‘𝑃))
8		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑐𝐶
9		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑑𝐶
10		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑎𝑑
11		nfcsb1v 3898	. . . . . 6 ⊢ Ⅎ𝑎⦋𝑐 / 𝑎⦌𝐶
12	10, 11	nfcsbw 3900	. . . . 5 ⊢ Ⅎ𝑎⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
13		nfcsb1v 3898	. . . . 5 ⊢ Ⅎ𝑏⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
14		csbeq1a 3888	. . . . . 6 ⊢ (𝑎 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑎⦌𝐶)
15		csbeq1a 3888	. . . . . 6 ⊢ (𝑏 = 𝑑 → ⦋𝑐 / 𝑎⦌𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
16	14, 15	sylan9eq 2790	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
17	8, 9, 12, 13, 16	cbvmpo 7501	. . . 4 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
18	17	oveqi 7418	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏)
19		eqidd 2736	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶))
20		equcom 2017	. . . . . . . 8 ⊢ (𝑎 = 𝑐 ↔ 𝑐 = 𝑎)
21		equcom 2017	. . . . . . . 8 ⊢ (𝑏 = 𝑑 ↔ 𝑑 = 𝑏)
22	20, 21	anbi12i 628	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) ↔ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏))
23	22, 16	sylbir 235	. . . . . 6 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
24	23	eqcomd 2741	. . . . 5 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
25	24	adantl 481	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏)) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
26		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
27		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
28		fvmpopr2d.3	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)
29	19, 25, 26, 27, 28	ovmpod 7559	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏) = 𝐶)
30	18, 29	eqtrid 2782	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
31	7, 30	eqtr3d 2772	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ⦋csb 3874  ⟨cop 4607  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410

This theorem is referenced by:  mpomulcn  24809  mnringmulrcld  44252