Theorem fnmpoovd 8066

Description: A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)

Hypotheses

Ref	Expression
fnmpoovd.m	⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
fnmpoovd.s	⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpoovd.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
fnmpoovd.c	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fnmpoovd	⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑖,𝑗   𝐵,𝑎,𝑏,𝑖,𝑗   𝐶,𝑖,𝑗   𝐷,𝑎,𝑏   𝑖,𝑀,𝑗   𝜑,𝑎,𝑏,𝑖,𝑗

Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑖,𝑗)   𝑈(𝑖,𝑗,𝑎,𝑏)   𝑀(𝑎,𝑏)   𝑉(𝑖,𝑗,𝑎,𝑏)

Proof of Theorem fnmpoovd

Step	Hyp	Ref	Expression
1		fnmpoovd.m	. . 3 ⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
2		fnmpoovd.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3	2	3expb 1133	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
4	3	ralrimivva 3205	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉)
5		eqid 2762	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
6	5	fnmpo 8050	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
8		eqfnov2 7526	. . 3 ⊢ ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
9	1, 7, 8	syl2anc 593	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
10		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑎𝐷
11		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑏𝐷
12		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑖𝐶
13		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
14		fnmpoovd.s	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
15	10, 11, 12, 13, 14	cbvmpo 7490	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
16	15	eqcomi 2771	. . . . . 6 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷))
18	17	oveqd 7413	. . . 4 ⊢ (𝜑 → (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗))
19	18	eqeq2d 2773	. . 3 ⊢ (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
20	19	2ralbidv 3226	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
21		simprl 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑖 ∈ 𝐴)
22		simprr 782	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑗 ∈ 𝐵)
23		fnmpoovd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
24	23	3expb 1133	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝐷 ∈ 𝑈)
25		eqid 2762	. . . . . 6 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
26	25	ovmpt4g 7543	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵 ∧ 𝐷 ∈ 𝑈) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
27	21, 22, 24, 26	syl3anc 1390	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
28	27	eqeq2d 2773	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29	28	2ralbidva 3224	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
30	9, 20, 29	3bitrd 307	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   × cxp 5645   Fn wfn 6516  (class class class)co 7396   ∈ cmpo 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971

This theorem is referenced by:  mpofrlmd  21829  fedgmullem2  33927