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Theorem fparlem3 7999

Description: Lemma for fpar 8001. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
fparlem3	⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹

Proof of Theorem fparlem3 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coiun 6181	. 2 ⊢ (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
2		inss1 4173	. . . . 5 ⊢ (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ dom 𝐹
3		fndm 6574	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	2, 3	sseqtrid 3983	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴)
5		dfco2a 6171	. . . 4 ⊢ ((dom 𝐹 ∩ ran (1^st ↾ (V × V))) ⊆ 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
6	4, 5	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∘ (1^st ↾ (V × V))) = ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
7	6	coeq2d 5791	. 2 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = (^◡(1^st ↾ (V × V)) ∘ ∪ 𝑥 ∈ 𝐴 ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
8		inss1 4173	. . . . . . . . 9 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ dom ({(𝐹‘𝑥)} × ({𝑥} × V))
9		dmxpss 6096	. . . . . . . . 9 ⊢ dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ⊆ {(𝐹‘𝑥)}
10	8, 9	sstri 3940	. . . . . . . 8 ⊢ (dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)}
11		dfco2a 6171	. . . . . . . 8 ⊢ ((dom ({(𝐹‘𝑥)} × ({𝑥} × V)) ∩ ran (1^st ↾ (V × V))) ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})))
12	10, 11	ax-mp 5	. . . . . . 7 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}))
13		fvex 6824	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
14		fparlem1 7997	. . . . . . . . . 10 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({𝑦} × V)
15		sneq 4581	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → {𝑦} = {(𝐹‘𝑥)})
16	15	xpeq1d 5636	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ({𝑦} × V) = ({(𝐹‘𝑥)} × V))
17	14, 16	eqtrid 2789	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (^◡(1^st ↾ (V × V)) “ {𝑦}) = ({(𝐹‘𝑥)} × V))
18	15	imaeq2d 5986	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}))
19		df-ima 5620	. . . . . . . . . . 11 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)})
20		ssid 3953	. . . . . . . . . . . . . 14 ⊢ {(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)}
21		xpssres 5947	. . . . . . . . . . . . . 14 ⊢ ({(𝐹‘𝑥)} ⊆ {(𝐹‘𝑥)} → (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V)))
22	20, 21	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({(𝐹‘𝑥)} × ({𝑥} × V))
23	22	rneqi 5865	. . . . . . . . . . . 12 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ran ({(𝐹‘𝑥)} × ({𝑥} × V))
24	13	snnz 4722	. . . . . . . . . . . . 13 ⊢ {(𝐹‘𝑥)} ≠ ∅
25		rnxp 6095	. . . . . . . . . . . . 13 ⊢ ({(𝐹‘𝑥)} ≠ ∅ → ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V))
26	24, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran ({(𝐹‘𝑥)} × ({𝑥} × V)) = ({𝑥} × V)
27	23, 26	eqtri 2765	. . . . . . . . . . 11 ⊢ ran (({(𝐹‘𝑥)} × ({𝑥} × V)) ↾ {(𝐹‘𝑥)}) = ({𝑥} × V)
28	19, 27	eqtri 2765	. . . . . . . . . 10 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {(𝐹‘𝑥)}) = ({𝑥} × V)
29	18, 28	eqtrdi 2793	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦}) = ({𝑥} × V))
30	17, 29	xpeq12d 5638	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V)))
31	13, 30	iunxsn 5033	. . . . . . 7 ⊢ ∪ 𝑦 ∈ {(𝐹‘𝑥)} ((^◡(1^st ↾ (V × V)) “ {𝑦}) × (({(𝐹‘𝑥)} × ({𝑥} × V)) “ {𝑦})) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
32	12, 31	eqtri 2765	. . . . . 6 ⊢ (({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (({(𝐹‘𝑥)} × V) × ({𝑥} × V))
33	32	cnveqi 5803	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V))
34		cnvco 5814	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × ({𝑥} × V)) ∘ (1^st ↾ (V × V))) = (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V)))
35		cnvxp 6082	. . . . 5 ⊢ ^◡(({(𝐹‘𝑥)} × V) × ({𝑥} × V)) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
36	33, 34, 35	3eqtr3i 2773	. . . 4 ⊢ (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (({𝑥} × V) × ({(𝐹‘𝑥)} × V))
37		fparlem1 7997	. . . . . . . . 9 ⊢ (^◡(1^st ↾ (V × V)) “ {𝑥}) = ({𝑥} × V)
38	37	xpeq2i 5634	. . . . . . . 8 ⊢ ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ({(𝐹‘𝑥)} × ({𝑥} × V))
39		fnsnfv 6886	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → {(𝐹‘𝑥)} = (𝐹 “ {𝑥}))
40	39	xpeq1d 5636	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
41	38, 40	eqtr3id 2791	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ({(𝐹‘𝑥)} × ({𝑥} × V)) = ((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
42	41	cnveqd 5804	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})))
43		cnvxp 6082	. . . . . 6 ⊢ ^◡((𝐹 “ {𝑥}) × (^◡(1^st ↾ (V × V)) “ {𝑥})) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))
44	42, 43	eqtrdi 2793	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ^◡({(𝐹‘𝑥)} × ({𝑥} × V)) = ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥})))
45	44	coeq2d 5791	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (^◡(1^st ↾ (V × V)) ∘ ^◡({(𝐹‘𝑥)} × ({𝑥} × V))) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
46	36, 45	eqtr3id 2791	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
47	46	iuneq2dv 4961	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)) = ∪ 𝑥 ∈ 𝐴 (^◡(1^st ↾ (V × V)) ∘ ((^◡(1^st ↾ (V × V)) “ {𝑥}) × (𝐹 “ {𝑥}))))
48	1, 7, 47	3eqtr4a 2803	1 ⊢ (𝐹 Fn 𝐴 → (^◡(1^st ↾ (V × V)) ∘ (𝐹 ∘ (1^st ↾ (V × V)))) = ∪ 𝑥 ∈ 𝐴 (({𝑥} × V) × ({(𝐹‘𝑥)} × V)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2941 Vcvv 3441 ∩ cin 3896 ⊆ wss 3897 ∅c0 4267 {csn 4571 ∪ ciun 4937 × cxp 5605 ^◡ccnv 5606 dom cdm 5607 ran crn 5608 ↾ cres 5609 “ cima 5610 ∘ ccom 5611 Fn wfn 6460 ‘cfv 6465 1^st c1st 7874

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5238 ax-nul 5245 ax-pr 5367 ax-un 7628

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4268 df-if 4472 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-iun 4939 df-br 5088 df-opab 5150 df-mpt 5171 df-id 5507 df-xp 5613 df-rel 5614 df-cnv 5615 df-co 5616 df-dm 5617 df-rn 5618 df-res 5619 df-ima 5620 df-iota 6417 df-fun 6467 df-fn 6468 df-f 6469 df-fv 6473 df-1st 7876 df-2nd 7877

This theorem is referenced by: fpar 8001

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