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Mirrors > Home > MPE Home > Th. List > fvunsn | Structured version Visualization version GIF version |
Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
fvunsn | ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundir 5871 | . . . 4 ⊢ ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) | |
2 | nelsn 4608 | . . . . . . 7 ⊢ (𝐵 ≠ 𝐷 → ¬ 𝐵 ∈ {𝐷}) | |
3 | ressnop0 6918 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ {𝐷} → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝐵 ≠ 𝐷 → ({〈𝐵, 𝐶〉} ↾ {𝐷}) = ∅) |
5 | 4 | uneq2d 4142 | . . . . 5 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = ((𝐴 ↾ {𝐷}) ∪ ∅)) |
6 | un0 4347 | . . . . 5 ⊢ ((𝐴 ↾ {𝐷}) ∪ ∅) = (𝐴 ↾ {𝐷}) | |
7 | 5, 6 | syl6eq 2875 | . . . 4 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ↾ {𝐷}) ∪ ({〈𝐵, 𝐶〉} ↾ {𝐷})) = (𝐴 ↾ {𝐷})) |
8 | 1, 7 | syl5eq 2871 | . . 3 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷}) = (𝐴 ↾ {𝐷})) |
9 | 8 | fveq1d 6675 | . 2 ⊢ (𝐵 ≠ 𝐷 → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ↾ {𝐷})‘𝐷)) |
10 | fvressn 6927 | . . 3 ⊢ (𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) | |
11 | fvprc 6666 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ∅) | |
12 | fvprc 6666 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = ∅) | |
13 | 11, 12 | eqtr4d 2862 | . . 3 ⊢ (¬ 𝐷 ∈ V → (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷)) |
14 | 10, 13 | pm2.61i 184 | . 2 ⊢ (((𝐴 ∪ {〈𝐵, 𝐶〉}) ↾ {𝐷})‘𝐷) = ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) |
15 | fvressn 6927 | . . 3 ⊢ (𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) | |
16 | fvprc 6666 | . . . 4 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = ∅) | |
17 | fvprc 6666 | . . . 4 ⊢ (¬ 𝐷 ∈ V → (𝐴‘𝐷) = ∅) | |
18 | 16, 17 | eqtr4d 2862 | . . 3 ⊢ (¬ 𝐷 ∈ V → ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷)) |
19 | 15, 18 | pm2.61i 184 | . 2 ⊢ ((𝐴 ↾ {𝐷})‘𝐷) = (𝐴‘𝐷) |
20 | 9, 14, 19 | 3eqtr3g 2882 | 1 ⊢ (𝐵 ≠ 𝐷 → ((𝐴 ∪ {〈𝐵, 𝐶〉})‘𝐷) = (𝐴‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∪ cun 3937 ∅c0 4294 {csn 4570 〈cop 4576 ↾ cres 5560 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-xp 5564 df-res 5570 df-iota 6317 df-fv 6366 |
This theorem is referenced by: fvpr1 6955 fvpr1g 6957 fvpr2g 6958 fvtp1 6960 fvtp1g 6963 ac6sfi 8765 cats1un 14086 ruclem6 15591 ruclem7 15592 wlkp1lem5 27462 wlkp1lem6 27463 fnchoice 41292 nnsum4primeseven 43972 nnsum4primesevenALTV 43973 |
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