Theorem fvmpopr2d 6158

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 6021	. . 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 1044	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶))
4		fvmpopr2d.2	. . . . 5 ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩)
5	4	3ad2ant1 1044	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑃 = ⟨𝑎, 𝑏⟩)
6	3, 5	fveq12d 5646	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = ((𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)‘⟨𝑎, 𝑏⟩))
7	1, 6	eqtr4id 2283	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝐹‘𝑃))
8		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑐𝐶
9		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑑𝐶
10		nfcv 2374	. . . . . 6 ⊢ Ⅎ𝑎𝑑
11		nfcsb1v 3160	. . . . . 6 ⊢ Ⅎ𝑎⦋𝑐 / 𝑎⦌𝐶
12	10, 11	nfcsbw 3164	. . . . 5 ⊢ Ⅎ𝑎⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
13		nfcsb1v 3160	. . . . 5 ⊢ Ⅎ𝑏⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶
14		csbeq1a 3136	. . . . . 6 ⊢ (𝑎 = 𝑐 → 𝐶 = ⦋𝑐 / 𝑎⦌𝐶)
15		csbeq1a 3136	. . . . . 6 ⊢ (𝑏 = 𝑑 → ⦋𝑐 / 𝑎⦌𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
16	14, 15	sylan9eq 2284	. . . . 5 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
17	8, 9, 12, 13, 16	cbvmpo 6100	. . . 4 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
18	17	oveqi 6031	. . 3 ⊢ (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏)
19		eqidd 2232	. . . 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 1754	. . . . . . . 8 ⊢ (𝑎 = 𝑐 ↔ 𝑐 = 𝑎)
21		equcom 1754	. . . . . . . 8 ⊢ (𝑏 = 𝑑 ↔ 𝑑 = 𝑏)
22	20, 21	anbi12i 460	. . . . . . 7 ⊢ ((𝑎 = 𝑐 ∧ 𝑏 = 𝑑) ↔ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏))
23	22, 16	sylbir 135	. . . . . 6 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → 𝐶 = ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)
24	23	eqcomd 2237	. . . . 5 ⊢ ((𝑐 = 𝑎 ∧ 𝑑 = 𝑏) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
25	24	adantl 277	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 = 𝑎 ∧ 𝑑 = 𝑏)) → ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶 = 𝐶)
26		simp2 1024	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
27		simp3 1025	. . . 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 6149	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑐 ∈ 𝐴, 𝑑 ∈ 𝐵 ↦ ⦋𝑑 / 𝑏⦌⦋𝑐 / 𝑎⦌𝐶)𝑏) = 𝐶)
30	18, 29	eqtrid 2276	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝑎(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
31	7, 30	eqtr3d 2266	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ⦋csb 3127  ⟨cop 3672  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023

This theorem is referenced by:  mpomulcn  15293