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Theorem fvmptssdm 5602
Description: If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping at a value in the domain of the mapping. (Contributed by Jim Kingdon, 3-Jan-2018.)
Hypothesis
Ref Expression
fvmpt2.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptssdm ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptssdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fveq2 5517 . . . . . 6 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
21sseq1d 3186 . . . . 5 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
32imbi2d 230 . . . 4 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
4 nfrab1 2657 . . . . . . 7 𝑥{𝑥𝐴𝐵 ∈ V}
54nfcri 2313 . . . . . 6 𝑥 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}
6 nfra1 2508 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
7 fvmpt2.1 . . . . . . . . . 10 𝐹 = (𝑥𝐴𝐵)
8 nfmpt1 4098 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
97, 8nfcxfr 2316 . . . . . . . . 9 𝑥𝐹
10 nfcv 2319 . . . . . . . . 9 𝑥𝑦
119, 10nffv 5527 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2319 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3150 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
146, 13nfim 1572 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
155, 14nfim 1572 . . . . 5 𝑥(𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
16 eleq1 2240 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} ↔ 𝑦 ∈ {𝑥𝐴𝐵 ∈ V}))
17 fveq2 5517 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1817sseq1d 3186 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1918imbi2d 230 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
2016, 19imbi12d 234 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶)) ↔ (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))))
217dmmpt 5126 . . . . . . 7 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
2221eleq2i 2244 . . . . . 6 (𝑥 ∈ dom 𝐹𝑥 ∈ {𝑥𝐴𝐵 ∈ V})
2321rabeq2i 2736 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
247fvmpt2 5601 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
25 eqimss 3211 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2624, 25syl 14 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2723, 26sylbi 121 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2827adantr 276 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐵)
297dmmptss 5127 . . . . . . . . . 10 dom 𝐹𝐴
3029sseli 3153 . . . . . . . . 9 (𝑥 ∈ dom 𝐹𝑥𝐴)
31 rsp 2524 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
3230, 31mpan9 281 . . . . . . . 8 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3328, 32sstrd 3167 . . . . . . 7 ((𝑥 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3433ex 115 . . . . . 6 (𝑥 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3522, 34sylbir 135 . . . . 5 (𝑥 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
3615, 20, 35chvar 1757 . . . 4 (𝑦 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
373, 36vtoclga 2805 . . 3 (𝐷 ∈ {𝑥𝐴𝐵 ∈ V} → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3837, 21eleq2s 2272 . 2 (𝐷 ∈ dom 𝐹 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3938imp 124 1 ((𝐷 ∈ dom 𝐹 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  {crab 2459  Vcvv 2739  wss 3131  cmpt 4066  dom cdm 4628  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fv 5226
This theorem is referenced by: (None)
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