Theorem fimaproj 8119

Description: Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)

Hypotheses

Ref	Expression
fvproj.h	⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
fimaproj.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
fimaproj.g	⊢ (𝜑 → 𝐺 Fn 𝐵)
fimaproj.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
fimaproj.y	⊢ (𝜑 → 𝑌 ⊆ 𝐵)

Assertion

Ref	Expression
fimaproj	⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem fimaproj

Dummy variables 𝑎 𝑏 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5436	. . . . 5 ⊢ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ V
2		fvproj.h	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
3		vex 3461	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
4		vex 3461	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	3, 4	op1std 7984	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
6	5	fveq2d 6875	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st ‘𝑧)) = (𝐹‘𝑥))
7	3, 4	op2ndd 7985	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
8	7	fveq2d 6875	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2nd ‘𝑧)) = (𝐺‘𝑦))
9	6, 8	opeq12d 4842	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
10	9	mpompt 7514	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
11	2, 10	eqtr4i 2791	. . . . 5 ⊢ 𝐻 = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
12	1, 11	fnmpti 6668	. . . 4 ⊢ 𝐻 Fn (𝐴 × 𝐵)
13		fimaproj.x	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
14		fimaproj.y	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝐵)
15		xpss12 5667	. . . . 5 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐵) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
16	13, 14, 15	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
17		fvelimab 6943	. . . 4 ⊢ ((𝐻 Fn (𝐴 × 𝐵) ∧ (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵)) → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
18	12, 16, 17	sylancr 598	. . 3 ⊢ (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
19		simp-4r 795	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑎 ∈ 𝑋)
20		simplr 780	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑏 ∈ 𝑌)
21		opelxpi 5689	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑌) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
22	19, 20, 21	syl2anc 595	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌))
23		simpllr 787	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐹‘𝑎) = (1st ‘𝑐))
24		simpr 489	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐺‘𝑏) = (2nd ‘𝑐))
25	23, 24	opeq12d 4842	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩ = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
26	13	ad5antr 746	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑋 ⊆ 𝐴)
27	26, 19	sseldd 3940	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑎 ∈ 𝐴)
28	14	ad5antr 746	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑌 ⊆ 𝐵)
29	28, 20	sseldd 3940	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑏 ∈ 𝐵)
30	2, 27, 29	fvproj 8118	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = ⟨(𝐹‘𝑎), (𝐺‘𝑏)⟩)
31		1st2nd2 8013	. . . . . . . . 9 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → 𝑐 = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
32	31	ad5antlr 747	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → 𝑐 = ⟨(1st ‘𝑐), (2nd ‘𝑐)⟩)
33	25, 30, 32	3eqtr4d 2810	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐)
34		fveqeq2 6880	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑎, 𝑏⟩ → ((𝐻‘𝑧) = 𝑐 ↔ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐))
35	34	rspcev 3584	. . . . . . 7 ⊢ ((⟨𝑎, 𝑏⟩ ∈ (𝑋 × 𝑌) ∧ (𝐻‘⟨𝑎, 𝑏⟩) = 𝑐) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
36	22, 33, 35	syl2anc 595	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) ∧ 𝑏 ∈ 𝑌) ∧ (𝐺‘𝑏) = (2nd ‘𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
37		fimaproj.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 Fn 𝐵)
38	37	ad3antrrr 742	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → 𝐺 Fn 𝐵)
39		fnfun 6625	. . . . . . . 8 ⊢ (𝐺 Fn 𝐵 → Fun 𝐺)
40	38, 39	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → Fun 𝐺)
41		xp2nd 8007	. . . . . . . 8 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → (2nd ‘𝑐) ∈ (𝐺 “ 𝑌))
42	41	ad3antlr 743	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → (2nd ‘𝑐) ∈ (𝐺 “ 𝑌))
43		fvelima 6936	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ (2nd ‘𝑐) ∈ (𝐺 “ 𝑌)) → ∃𝑏 ∈ 𝑌 (𝐺‘𝑏) = (2nd ‘𝑐))
44	40, 42, 43	syl2anc 595	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → ∃𝑏 ∈ 𝑌 (𝐺‘𝑏) = (2nd ‘𝑐))
45	36, 44	r19.29a 3173	. . . . 5 ⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = (1st ‘𝑐)) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
46		fimaproj.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝐴)
47	46	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → 𝐹 Fn 𝐴)
48		fnfun 6625	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
49	47, 48	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → Fun 𝐹)
50		xp1st 8006	. . . . . . 7 ⊢ (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) → (1st ‘𝑐) ∈ (𝐹 “ 𝑋))
51	50	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → (1st ‘𝑐) ∈ (𝐹 “ 𝑋))
52		fvelima 6936	. . . . . 6 ⊢ ((Fun 𝐹 ∧ (1st ‘𝑐) ∈ (𝐹 “ 𝑋)) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = (1st ‘𝑐))
53	49, 51, 52	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = (1st ‘𝑐))
54	45, 53	r19.29a 3173	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) → ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐)
55		simpr 489	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) = 𝑐)
56	16	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝑋 × 𝑌) ⊆ (𝐴 × 𝐵))
57		simplr 780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑧 ∈ (𝑋 × 𝑌))
58	56, 57	sseldd 3940	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑧 ∈ (𝐴 × 𝐵))
59	11	fvmpt2 6991	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ V) → (𝐻‘𝑧) = ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
60	58, 1, 59	sylancl 597	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) = ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩)
61	46	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝐹 Fn 𝐴)
62	13	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑋 ⊆ 𝐴)
63		xp1st 8006	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
64	57, 63	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (1st ‘𝑧) ∈ 𝑋)
65		fnfvima 7221	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ∧ (1st ‘𝑧) ∈ 𝑋) → (𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋))
66	61, 62, 64, 65	syl3anc 1394	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋))
67	37	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝐺 Fn 𝐵)
68	14	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑌 ⊆ 𝐵)
69		xp2nd 8007	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2nd ‘𝑧) ∈ 𝑌)
70	57, 69	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (2nd ‘𝑧) ∈ 𝑌)
71		fnfvima 7221	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑌 ⊆ 𝐵 ∧ (2nd ‘𝑧) ∈ 𝑌) → (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌))
72	67, 68, 70, 71	syl3anc 1394	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌))
73		opelxpi 5689	. . . . . . . 8 ⊢ (((𝐹‘(1st ‘𝑧)) ∈ (𝐹 “ 𝑋) ∧ (𝐺‘(2nd ‘𝑧)) ∈ (𝐺 “ 𝑌)) → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
74	66, 72, 73	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → ⟨(𝐹‘(1st ‘𝑧)), (𝐺‘(2nd ‘𝑧))⟩ ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
75	60, 74	eqeltrd 2865	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → (𝐻‘𝑧) ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
76	55, 75	eqeltrrd 2866	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ (𝑋 × 𝑌)) ∧ (𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
77	76	r19.29an 3169	. . . 4 ⊢ ((𝜑 ∧ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐) → 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
78	54, 77	impbida 812	. . 3 ⊢ (𝜑 → (𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)) ↔ ∃𝑧 ∈ (𝑋 × 𝑌)(𝐻‘𝑧) = 𝑐))
79	18, 78	bitr4d 285	. 2 ⊢ (𝜑 → (𝑐 ∈ (𝐻 “ (𝑋 × 𝑌)) ↔ 𝑐 ∈ ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))))
80	79	eqrdv 2763	1 ⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ⟨cop 4591   ↦ cmpt 5186   × cxp 5650   “ cima 5655  Fun wfun 6519   Fn wfn 6520  ‘cfv 6525   ∈ cmpo 7402  1^st c1st 7972  2^nd c2nd 7973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975

This theorem is referenced by:  txomap  34141