Theorem dmmpossx 8048

Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

Ref	Expression
fmpox.1	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
dmmpossx	⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dmmpossx

Dummy variables 𝑢 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑢𝐵
2		nfcsb1v 3889	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
3		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcsb1v 3889	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
6		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦𝑢
7		nfcsb1v 3889	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
8	6, 7	nfcsbw 3891	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
9		csbeq1a 3879	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
10		csbeq1a 3879	. . . . . 6 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11		csbeq1a 3879	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	10, 11	sylan9eqr 2787	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 7485	. . . 4 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
14		fmpox.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
15		vex 3454	. . . . . . . 8 ⊢ 𝑢 ∈ V
16		vex 3454	. . . . . . . 8 ⊢ 𝑣 ∈ V
17	15, 16	op1std 7981	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑡) = 𝑢)
18	17	csbeq1d 3869	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
19	15, 16	op2ndd 7982	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑡) = 𝑣)
20	19	csbeq1d 3869	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
21	20	csbeq2dv 3872	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
22	18, 21	eqtrd 2765	. . . . 5 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
23	22	mpomptx 7505	. . . 4 ⊢ (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
24	13, 14, 23	3eqtr4i 2763	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
25	24	dmmptss 6217	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
26		nfcv 2892	. . 3 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
27		nfcv 2892	. . . 4 ⊢ Ⅎ𝑥{𝑢}
28	27, 2	nfxp 5674	. . 3 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
29		sneq 4602	. . . 4 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
30	29, 9	xpeq12d 5672	. . 3 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
31	26, 28, 30	cbviun 5003	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
32	25, 31	sseqtrri 3999	1 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

Syntax hints:   = wceq 1540  ⦋csb 3865   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ ciun 4958   ↦ cmpt 5191   × cxp 5639  dom cdm 5641  ‘cfv 6514   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972

This theorem is referenced by:  mpoexxg  8057  mpoxeldm  8193  mpoxopn0yelv  8195  mpoxopxnop0  8197  dmcoass  18035  ply1frcl  22212  dvbsss  25810  perfdvf  25811