Theorem fnmpoovd 6273

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 1206	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
4	3	ralrimivva 2579	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉)
5		eqid 2196	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
6	5	fnmpo 6260	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
7	4, 6	syl 14	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
8		eqfnov2 6030	. . 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 411	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
10		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑎𝐷
11		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑏𝐷
12		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑖𝐶
13		nfcv 2339	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
14		fnmpoovd.s	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
15	10, 11, 12, 13, 14	cbvmpo 6001	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
16	15	eqcomi 2200	. . . . . 6 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
17	16	a1i 9	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷))
18	17	oveqd 5939	. . . 4 ⊢ (𝜑 → (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗))
19	18	eqeq2d 2208	. . 3 ⊢ (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
20	19	2ralbidv 2521	. 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 529	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑖 ∈ 𝐴)
22		simprr 531	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑗 ∈ 𝐵)
23		fnmpoovd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
24	23	3expb 1206	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝐷 ∈ 𝑈)
25		eqid 2196	. . . . . 6 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
26	25	ovmpt4g 6045	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵 ∧ 𝐷 ∈ 𝑈) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
27	21, 22, 24, 26	syl3anc 1249	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
28	27	eqeq2d 2208	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29	28	2ralbidva 2519	. 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 214	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475   × cxp 4661   Fn wfn 5253  (class class class)co 5922   ∈ cmpo 5924

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199

This theorem is referenced by: (None)