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Mirrors > Home > ILE Home > Th. List > fvsnun1 | Unicode version |
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5618. (Contributed by NM, 23-Sep-2007.) |
Ref | Expression |
---|---|
fvsnun.1 | |
fvsnun.2 | |
fvsnun.3 |
Ref | Expression |
---|---|
fvsnun1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvsnun.3 | . . . . 5 | |
2 | 1 | reseq1i 4815 | . . . 4 |
3 | resundir 4833 | . . . . 5 | |
4 | incom 3268 | . . . . . . . . 9 | |
5 | disjdif 3435 | . . . . . . . . 9 | |
6 | 4, 5 | eqtri 2160 | . . . . . . . 8 |
7 | resdisj 4967 | . . . . . . . 8 | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 |
9 | 8 | uneq2i 3227 | . . . . . 6 |
10 | un0 3396 | . . . . . 6 | |
11 | 9, 10 | eqtri 2160 | . . . . 5 |
12 | 3, 11 | eqtri 2160 | . . . 4 |
13 | 2, 12 | eqtri 2160 | . . 3 |
14 | 13 | fveq1i 5422 | . 2 |
15 | fvsnun.1 | . . . 4 | |
16 | 15 | snid 3556 | . . 3 |
17 | fvres 5445 | . . 3 | |
18 | 16, 17 | ax-mp 5 | . 2 |
19 | fvres 5445 | . . . 4 | |
20 | 16, 19 | ax-mp 5 | . . 3 |
21 | fvsnun.2 | . . . 4 | |
22 | 15, 21 | fvsn 5615 | . . 3 |
23 | 20, 22 | eqtri 2160 | . 2 |
24 | 14, 18, 23 | 3eqtr3i 2168 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 cvv 2686 cdif 3068 cun 3069 cin 3070 c0 3363 csn 3527 cop 3530 cres 4541 cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: fac0 10474 |
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