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Theorem fvmptss 6943
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 6179 . . . 4 dom 𝐹𝐴
32sseli 3928 . . 3 (𝐷 ∈ dom 𝐹𝐷𝐴)
4 fveq2 6825 . . . . . . 7 (𝑦 = 𝐷 → (𝐹𝑦) = (𝐹𝐷))
54sseq1d 3963 . . . . . 6 (𝑦 = 𝐷 → ((𝐹𝑦) ⊆ 𝐶 ↔ (𝐹𝐷) ⊆ 𝐶))
65imbi2d 340 . . . . 5 (𝑦 = 𝐷 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)))
7 nfcv 2904 . . . . . 6 𝑥𝑦
8 nfra1 3263 . . . . . . 7 𝑥𝑥𝐴 𝐵𝐶
9 nfmpt1 5200 . . . . . . . . . 10 𝑥(𝑥𝐴𝐵)
101, 9nfcxfr 2902 . . . . . . . . 9 𝑥𝐹
1110, 7nffv 6835 . . . . . . . 8 𝑥(𝐹𝑦)
12 nfcv 2904 . . . . . . . 8 𝑥𝐶
1311, 12nfss 3924 . . . . . . 7 𝑥(𝐹𝑦) ⊆ 𝐶
148, 13nfim 1898 . . . . . 6 𝑥(∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)
15 fveq2 6825 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1615sseq1d 3963 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ⊆ 𝐶 ↔ (𝐹𝑦) ⊆ 𝐶))
1716imbi2d 340 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶) ↔ (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶)))
181dmmpt 6178 . . . . . . . . . . 11 dom 𝐹 = {𝑥𝐴𝐵 ∈ V}
1918rabeq2i 3425 . . . . . . . . . 10 (𝑥 ∈ dom 𝐹 ↔ (𝑥𝐴𝐵 ∈ V))
201fvmpt2 6942 . . . . . . . . . . 11 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) = 𝐵)
21 eqimss 3988 . . . . . . . . . . 11 ((𝐹𝑥) = 𝐵 → (𝐹𝑥) ⊆ 𝐵)
2220, 21syl 17 . . . . . . . . . 10 ((𝑥𝐴𝐵 ∈ V) → (𝐹𝑥) ⊆ 𝐵)
2319, 22sylbi 216 . . . . . . . . 9 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
24 ndmfv 6860 . . . . . . . . . 10 𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ∅)
25 0ss 4343 . . . . . . . . . 10 ∅ ⊆ 𝐵
2624, 25eqsstrdi 3986 . . . . . . . . 9 𝑥 ∈ dom 𝐹 → (𝐹𝑥) ⊆ 𝐵)
2723, 26pm2.61i 182 . . . . . . . 8 (𝐹𝑥) ⊆ 𝐵
28 rsp 3226 . . . . . . . . 9 (∀𝑥𝐴 𝐵𝐶 → (𝑥𝐴𝐵𝐶))
2928impcom 408 . . . . . . . 8 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → 𝐵𝐶)
3027, 29sstrid 3943 . . . . . . 7 ((𝑥𝐴 ∧ ∀𝑥𝐴 𝐵𝐶) → (𝐹𝑥) ⊆ 𝐶)
3130ex 413 . . . . . 6 (𝑥𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑥) ⊆ 𝐶))
327, 14, 17, 31vtoclgaf 3521 . . . . 5 (𝑦𝐴 → (∀𝑥𝐴 𝐵𝐶 → (𝐹𝑦) ⊆ 𝐶))
336, 32vtoclga 3522 . . . 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 6860 . . . 4 𝐷 ∈ dom 𝐹 → (𝐹𝐷) = ∅)
3736adantl 482 . . 3 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) = ∅)
38 0ss 4343 . . 3 ∅ ⊆ 𝐶
3937, 38eqsstrdi 3986 . 2 ((∀𝑥𝐴 𝐵𝐶 ∧ ¬ 𝐷 ∈ dom 𝐹) → (𝐹𝐷) ⊆ 𝐶)
4035, 39pm2.61dan 810 1 (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  wss 3898  c0 4269  cmpt 5175  dom cdm 5620  cfv 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fv 6487
This theorem is referenced by:  relmptopab  7581  ovmptss  8001
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