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Theorem ovmptss 7303
Description: If all the values of the mapping are subsets of a class 𝑋, then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
ovmptss.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
ovmptss (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem ovmptss
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovmptss.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 mpt2mptsx 7278 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
31, 2eqtri 2673 . . 3 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
43fvmptss 6331 . 2 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5 vex 3234 . . . . . . . 8 𝑢 ∈ V
6 vex 3234 . . . . . . . 8 𝑣 ∈ V
75, 6op1std 7220 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
87csbeq1d 3573 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
95, 6op2ndd 7221 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
109csbeq1d 3573 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
1110csbeq2dv 4025 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
128, 11eqtrd 2685 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1312sseq1d 3665 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
1413raliunxp 5294 . . 3 (∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
15 nfcv 2793 . . . . 5 𝑢({𝑥} × 𝐵)
16 nfcv 2793 . . . . . 6 𝑥{𝑢}
17 nfcsb1v 3582 . . . . . 6 𝑥𝑢 / 𝑥𝐵
1816, 17nfxp 5176 . . . . 5 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
19 sneq 4220 . . . . . 6 (𝑥 = 𝑢 → {𝑥} = {𝑢})
20 csbeq1a 3575 . . . . . 6 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
2119, 20xpeq12d 5174 . . . . 5 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
2215, 18, 21cbviun 4589 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
2322raleqi 3172 . . 3 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
24 nfv 1883 . . . 4 𝑢𝑦𝐵 𝐶𝑋
25 nfcsb1v 3582 . . . . . 6 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
26 nfcv 2793 . . . . . 6 𝑥𝑋
2725, 26nfss 3629 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
2817, 27nfral 2974 . . . 4 𝑥𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
29 nfv 1883 . . . . . 6 𝑣 𝐶𝑋
30 nfcsb1v 3582 . . . . . . 7 𝑦𝑣 / 𝑦𝐶
31 nfcv 2793 . . . . . . 7 𝑦𝑋
3230, 31nfss 3629 . . . . . 6 𝑦𝑣 / 𝑦𝐶𝑋
33 csbeq1a 3575 . . . . . . 7 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
3433sseq1d 3665 . . . . . 6 (𝑦 = 𝑣 → (𝐶𝑋𝑣 / 𝑦𝐶𝑋))
3529, 32, 34cbvral 3197 . . . . 5 (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋)
36 csbeq1a 3575 . . . . . . 7 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3736sseq1d 3665 . . . . . 6 (𝑥 = 𝑢 → (𝑣 / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3820, 37raleqbidv 3182 . . . . 5 (𝑥 = 𝑢 → (∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3935, 38syl5bb 272 . . . 4 (𝑥 = 𝑢 → (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
4024, 28, 39cbvral 3197 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
4114, 23, 403bitr4ri 293 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
42 df-ov 6693 . . 3 (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
4342sseq1i 3662 . 2 ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
444, 41, 433imtr4i 281 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wral 2941  csb 3566  wss 3607  {csn 4210  cop 4216   ciun 4552  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211
This theorem is referenced by:  relmpt2opab  7304  relxpchom  16868  reldv  23679
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