Theorem fvmpopr2d 7503

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 7344	. . 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 6824	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩))
7	1, 6	eqtr4id 2785	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝐹‘𝑃))
8		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑐𝐶
9		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑑𝐶
10		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑎𝑑
11		nfcsb1v 3869	. . . . . 6 ⊢ Ⅎ𝑎⦋𝑐 / 𝑎⦌𝐶
12	10, 11	nfcsbw 3871	. . . . 5 ⊢ Ⅎ𝑎⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
13		nfcsb1v 3869	. . . . 5 ⊢ Ⅎ𝑏⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
14		csbeq1a 3859	. . . . . 6 ⊢ (𝑎 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑎⦌𝐶)
15		csbeq1a 3859	. . . . . 6 ⊢ (𝑏 = 𝑑 → ⦋𝑐 / 𝑎⦌𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
16	14, 15	sylan9eq 2786	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
17	8, 9, 12, 13, 16	cbvmpo 7435	. . . 4 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
18	17	oveqi 7354	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏)
19		eqidd 2732	. . . 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 2019	. . . . . . . 8 ⊢ (𝑎 = 𝑐 ↔ 𝑐 = 𝑎)
21		equcom 2019	. . . . . . . 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 2737	. . . . 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 7493	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏) = 𝐶)
30	18, 29	eqtrid 2778	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
31	7, 30	eqtr3d 2768	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ⦋csb 3845  ⟨cop 4577  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346

This theorem is referenced by:  mpomulcn  24780  mnringmulrcld  44261