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Theorem ovmptss 8079
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 mpomptsx 8050 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
31, 2eqtri 2761 . . 3 𝐹 = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
43fvmptss 7011 . 2 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 → (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
5 vex 3479 . . . . . . . 8 𝑢 ∈ V
6 vex 3479 . . . . . . . 8 𝑣 ∈ V
75, 6op1std 7985 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
87csbeq1d 3898 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
95, 6op2ndd 7986 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
109csbeq1d 3898 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
1110csbeq2dv 3901 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
128, 11eqtrd 2773 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
1312sseq1d 4014 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
1413raliunxp 5840 . . 3 (∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
15 nfcv 2904 . . . . 5 𝑢({𝑥} × 𝐵)
16 nfcv 2904 . . . . . 6 𝑥{𝑢}
17 nfcsb1v 3919 . . . . . 6 𝑥𝑢 / 𝑥𝐵
1816, 17nfxp 5710 . . . . 5 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
19 sneq 4639 . . . . . 6 (𝑥 = 𝑢 → {𝑥} = {𝑢})
20 csbeq1a 3908 . . . . . 6 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
2119, 20xpeq12d 5708 . . . . 5 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
2215, 18, 21cbviun 5040 . . . 4 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
2322raleqi 3324 . . 3 (∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋 ↔ ∀𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
24 nfv 1918 . . . 4 𝑢𝑦𝐵 𝐶𝑋
25 nfcsb1v 3919 . . . . . 6 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
26 nfcv 2904 . . . . . 6 𝑥𝑋
2725, 26nfss 3975 . . . . 5 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
2817, 27nfralw 3309 . . . 4 𝑥𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋
29 nfv 1918 . . . . . 6 𝑣 𝐶𝑋
30 nfcsb1v 3919 . . . . . . 7 𝑦𝑣 / 𝑦𝐶
31 nfcv 2904 . . . . . . 7 𝑦𝑋
3230, 31nfss 3975 . . . . . 6 𝑦𝑣 / 𝑦𝐶𝑋
33 csbeq1a 3908 . . . . . . 7 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
3433sseq1d 4014 . . . . . 6 (𝑦 = 𝑣 → (𝐶𝑋𝑣 / 𝑦𝐶𝑋))
3529, 32, 34cbvralw 3304 . . . . 5 (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋)
36 csbeq1a 3908 . . . . . . 7 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3736sseq1d 4014 . . . . . 6 (𝑥 = 𝑢 → (𝑣 / 𝑦𝐶𝑋𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3820, 37raleqbidv 3343 . . . . 5 (𝑥 = 𝑢 → (∀𝑣𝐵 𝑣 / 𝑦𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
3935, 38bitrid 283 . . . 4 (𝑥 = 𝑢 → (∀𝑦𝐵 𝐶𝑋 ↔ ∀𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋))
4024, 28, 39cbvralw 3304 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑢𝐴𝑣 𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶𝑋)
4114, 23, 403bitr4ri 304 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 ↔ ∀𝑧 𝑥𝐴 ({𝑥} × 𝐵)(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑋)
42 df-ov 7412 . . 3 (𝐸𝐹𝐺) = (𝐹‘⟨𝐸, 𝐺⟩)
4342sseq1i 4011 . 2 ((𝐸𝐹𝐺) ⊆ 𝑋 ↔ (𝐹‘⟨𝐸, 𝐺⟩) ⊆ 𝑋)
444, 41, 433imtr4i 292 1 (∀𝑥𝐴𝑦𝐵 𝐶𝑋 → (𝐸𝐹𝐺) ⊆ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wral 3062  csb 3894  wss 3949  {csn 4629  cop 4635   ciun 4998  cmpt 5232   × cxp 5675  cfv 6544  (class class class)co 7409  cmpo 7411  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976
This theorem is referenced by:  relmpoopab  8080  relxpchom  18133  reldv  25387
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