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Mirrors > Home > MPE Home > Th. List > fvsnun1 | Structured version Visualization version Unicode version |
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6449. (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 5392 | . . . 4 |
3 | resundir 5411 | . . . . 5 | |
4 | incom 3805 | . . . . . . . . 9 | |
5 | disjdif 4040 | . . . . . . . . 9 | |
6 | 4, 5 | eqtri 2644 | . . . . . . . 8 |
7 | resdisj 5563 | . . . . . . . 8 | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 |
9 | 8 | uneq2i 3764 | . . . . . 6 |
10 | un0 3967 | . . . . . 6 | |
11 | 9, 10 | eqtri 2644 | . . . . 5 |
12 | 3, 11 | eqtri 2644 | . . . 4 |
13 | 2, 12 | eqtri 2644 | . . 3 |
14 | 13 | fveq1i 6192 | . 2 |
15 | fvsnun.1 | . . . 4 | |
16 | 15 | snid 4208 | . . 3 |
17 | fvres 6207 | . . 3 | |
18 | 16, 17 | ax-mp 5 | . 2 |
19 | fvres 6207 | . . . 4 | |
20 | 16, 19 | ax-mp 5 | . . 3 |
21 | fvsnun.2 | . . . 4 | |
22 | 15, 21 | fvsn 6446 | . . 3 |
23 | 20, 22 | eqtri 2644 | . 2 |
24 | 14, 18, 23 | 3eqtr3i 2652 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 cvv 3200 cdif 3571 cun 3572 cin 3573 c0 3915 csn 4177 cop 4183 cres 5116 cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: fac0 13063 ruclem4 14963 |
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