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Theorem fvmptss 6960
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 6193 . . . 4 dom 𝐹𝐴
32sseli 3940 . . 3 (𝐷 ∈ dom 𝐹𝐷𝐴)
4 fveq2 6842 . . . . . . 7 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
54sseq1d 3975 . . . . . 6 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
65imbi2d 340 . . . . 5 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
7 nfcv 2907 . . . . . 6 𝑥𝑦
8 nfra1 3267 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
9 nfmpt1 5213 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
101, 9nfcxfr 2905 . . . . . . . . 9 𝑥𝐹
1110, 7nffv 6852 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2907 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3936 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
148, 13nfim 1899 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
15 fveq2 6842 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1615sseq1d 3975 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1716imbi2d 340 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
181dmmpt 6192 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
1918reqabi 3429 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
201fvmpt2 6959 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
21 eqimss 4000 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2220, 21syl 17 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2319, 22sylbi 216 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
24 ndmfv 6877 . . . . . . . . . 10 𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ∅)
25 0ss 4356 . . . . . . . . . 10 ∅ ⊆ 𝐵
2624, 25eqsstrdi 3998 . . . . . . . . 9 𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2723, 26pm2.61i 182 . . . . . . . 8 (𝐹𝑥) ⊆ 𝐵
28 rsp 3230 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
2928impcom 408 . . . . . . . 8 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3027, 29sstrid 3955 . . . . . . 7 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3130ex 413 . . . . . 6 (𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
327, 14, 17, 31vtoclgaf 3533 . . . . 5 (𝑦𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
336, 32vtoclga 3534 . . . 4 (𝐷𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶))
3433impcom 408 . . 3 ((∀𝑥𝐴 𝐵𝐶𝐷𝐴) → (𝐹𝐷) ⊆ 𝐶)
353, 34sylan2 593 . 2 ((∀𝑥𝐴 𝐵𝐶𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
36 ndmfv 6877 . . . 4 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
3736adantl 482 . . 3 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) = ∅)
38 0ss 4356 . . 3 ∅ ⊆ 𝐶
3937, 38eqsstrdi 3998 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
4035, 39pm2.61dan 811 1 (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  c0 4282  cmpt 5188  dom cdm 5633  cfv 6496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504
This theorem is referenced by:  relmptopab  7603  ovmptss  8025
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