Theorem dmmpossx 8077

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 2923	. . . . 5 ⊢ Ⅎ𝑢𝐵
2		nfcsb1v 3871	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
3		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑢𝐶
4		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑣𝐶
5		nfcsb1v 3871	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
6		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑦𝑢
7		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
8	6, 7	nfcsbw 3873	. . . . 5 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
9		csbeq1a 3861	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
10		csbeq1a 3861	. . . . . 6 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
11		csbeq1a 3861	. . . . . 6 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
12	10, 11	sylan9eqr 2818	. . . . 5 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpox 7513	. . . 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 3455	. . . . . . . 8 ⊢ 𝑢 ∈ V
16		vex 3455	. . . . . . . 8 ⊢ 𝑣 ∈ V
17	15, 16	op1std 8011	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑡) = 𝑢)
18	17	csbeq1d 3851	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
19	15, 16	op2ndd 8012	. . . . . . . 8 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑡) = 𝑣)
20	19	csbeq1d 3851	. . . . . . 7 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
21	20	csbeq2dv 3854	. . . . . 6 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
22	18, 21	eqtrd 2796	. . . . 5 ⊢ (𝑡 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
23	22	mpomptx 7533	. . . 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 2794	. . 3 ⊢ 𝐹 = (𝑡 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑡) / 𝑥⦌⦋(2nd ‘𝑡) / 𝑦⦌𝐶)
25	24	dmmptss 6242	. 2 ⊢ dom 𝐹 ⊆ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
26		nfcv 2923	. . 3 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
27		nfcv 2923	. . . 4 ⊢ Ⅎ𝑥{𝑢}
28	27, 2	nfxp 5684	. . 3 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
29		sneq 4594	. . . 4 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
30	29, 9	xpeq12d 5682	. . 3 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
31	26, 28, 30	cbviun 4993	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
32	25, 31	sseqtrri 3980	1 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)

This proof depends on syntax axioms:   = wceq 1570  ⦋csb 3847   ⊆ wss 3899  {csn 4584  ⟨cop 4590  ∪ ciun 4951   ↦ cmpt 5186   × cxp 5649  dom cdm 5651  ‘cfv 6538   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fv 6546  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  mpoexxg  8088  mpoxeldm  8228  mpoxopn0yelv  8230  mpoxopxnop0  8232  dmcoass  18241  ply1frcl  22636  dvbsss  26222  perfdvf  26223