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Theorem fvmptss 6259
Description: If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.)
Hypothesis
Ref Expression
mptrcl.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
fvmptss (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)   𝐹(𝑥)

Proof of Theorem fvmptss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
21dmmptss 5600 . . . 4 dom 𝐹𝐴
32sseli 3584 . . 3 (𝐷 ∈ dom 𝐹𝐷𝐴)
4 fveq2 6158 . . . . . . 7 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
54sseq1d 3617 . . . . . 6 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
65imbi2d 330 . . . . 5 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
7 nfcv 2761 . . . . . 6 𝑥𝑦
8 nfra1 2937 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
9 nfmpt1 4717 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
101, 9nfcxfr 2759 . . . . . . . . 9 𝑥𝐹
1110, 7nffv 6165 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2761 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3581 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
148, 13nfim 1822 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
15 fveq2 6158 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1615sseq1d 3617 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1716imbi2d 330 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
181dmmpt 5599 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
1918rabeq2i 3187 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
201fvmpt2 6258 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
21 eqimss 3642 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2220, 21syl 17 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2319, 22sylbi 207 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
24 ndmfv 6185 . . . . . . . . . 10 𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ∅)
25 0ss 3950 . . . . . . . . . 10 ∅ ⊆ 𝐵
2624, 25syl6eqss 3640 . . . . . . . . 9 𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2723, 26pm2.61i 176 . . . . . . . 8 (𝐹𝑥) ⊆ 𝐵
28 rsp 2925 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
2928impcom 446 . . . . . . . 8 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3027, 29syl5ss 3599 . . . . . . 7 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3130ex 450 . . . . . 6 (𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
327, 14, 17, 31vtoclgaf 3261 . . . . 5 (𝑦𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
336, 32vtoclga 3262 . . . 4 (𝐷𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3433impcom 446 . . 3 ((∀𝑥𝐴 𝐵𝐶𝐷𝐴) → (𝐹𝐷) ⊆ 𝐶)
353, 34sylan2 491 . 2 ((∀𝑥𝐴 𝐵𝐶𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
36 ndmfv 6185 . . . 4 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
3736adantl 482 . . 3 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) = ∅)
38 0ss 3950 . . 3 ∅ ⊆ 𝐶
3937, 38syl6eqss 3640 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
4035, 39pm2.61dan 831 1 (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  wss 3560  c0 3897  cmpt 4683  dom cdm 5084  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865
This theorem is referenced by:  relmptopab  6848  ovmptss  7218
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