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Mirrors > Home > MPE Home > Th. List > newval | Structured version Visualization version GIF version |
Description: The value of the new options function. (Contributed by Scott Fenton, 9-Oct-2024.) |
Ref | Expression |
---|---|
newval | ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M ‘𝑥) = ( M ‘𝐴)) | |
2 | fveq2 6907 | . . . 4 ⊢ (𝑥 = 𝐴 → ( O ‘𝑥) = ( O ‘𝐴)) | |
3 | 1, 2 | difeq12d 4137 | . . 3 ⊢ (𝑥 = 𝐴 → (( M ‘𝑥) ∖ ( O ‘𝑥)) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
4 | df-new 27903 | . . 3 ⊢ N = (𝑥 ∈ On ↦ (( M ‘𝑥) ∖ ( O ‘𝑥))) | |
5 | fvex 6920 | . . . 4 ⊢ ( M ‘𝐴) ∈ V | |
6 | 5 | difexi 5336 | . . 3 ⊢ (( M ‘𝐴) ∖ ( O ‘𝐴)) ∈ V |
7 | 3, 4, 6 | fvmpt 7016 | . 2 ⊢ (𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
8 | 4 | fvmptndm 7047 | . . 3 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = ∅) |
9 | df-made 27901 | . . . . . . . . 9 ⊢ M = recs((𝑓 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑓 × 𝒫 ∪ ran 𝑓)))) | |
10 | 9 | tfr1 8436 | . . . . . . . 8 ⊢ M Fn On |
11 | 10 | fndmi 6673 | . . . . . . 7 ⊢ dom M = On |
12 | 11 | eleq2i 2831 | . . . . . 6 ⊢ (𝐴 ∈ dom M ↔ 𝐴 ∈ On) |
13 | ndmfv 6942 | . . . . . 6 ⊢ (¬ 𝐴 ∈ dom M → ( M ‘𝐴) = ∅) | |
14 | 12, 13 | sylnbir 331 | . . . . 5 ⊢ (¬ 𝐴 ∈ On → ( M ‘𝐴) = ∅) |
15 | 14 | difeq1d 4135 | . . . 4 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = (∅ ∖ ( O ‘𝐴))) |
16 | 0dif 4411 | . . . 4 ⊢ (∅ ∖ ( O ‘𝐴)) = ∅ | |
17 | 15, 16 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐴 ∈ On → (( M ‘𝐴) ∖ ( O ‘𝐴)) = ∅) |
18 | 8, 17 | eqtr4d 2778 | . 2 ⊢ (¬ 𝐴 ∈ On → ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴))) |
19 | 7, 18 | pm2.61i 182 | 1 ⊢ ( N ‘𝐴) = (( M ‘𝐴) ∖ ( O ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ran crn 5690 “ cima 5692 Oncon0 6386 ‘cfv 6563 |s cscut 27842 M cmade 27896 O cold 27897 N cnew 27898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-made 27901 df-new 27903 |
This theorem is referenced by: new0 27928 madeun 27937 newbday 27955 |
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