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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsneq | Structured version Visualization version Unicode version |
Description: Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
fsneq.a | |
fsneq.b | |
fsneq.f | |
fsneq.g |
Ref | Expression |
---|---|
fsneq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsneq.f | . . 3 | |
2 | fsneq.g | . . 3 | |
3 | eqfnfv 6311 | . . 3 | |
4 | 1, 2, 3 | syl2anc 693 | . 2 |
5 | fsneq.a | . . . . . . . 8 | |
6 | snidg 4206 | . . . . . . . 8 | |
7 | 5, 6 | syl 17 | . . . . . . 7 |
8 | fsneq.b | . . . . . . . . 9 | |
9 | 8 | eqcomi 2631 | . . . . . . . 8 |
10 | 9 | a1i 11 | . . . . . . 7 |
11 | 7, 10 | eleqtrd 2703 | . . . . . 6 |
12 | 11 | adantr 481 | . . . . 5 |
13 | simpr 477 | . . . . 5 | |
14 | fveq2 6191 | . . . . . . 7 | |
15 | fveq2 6191 | . . . . . . 7 | |
16 | 14, 15 | eqeq12d 2637 | . . . . . 6 |
17 | 16 | rspcva 3307 | . . . . 5 |
18 | 12, 13, 17 | syl2anc 693 | . . . 4 |
19 | 18 | ex 450 | . . 3 |
20 | simpl 473 | . . . . . . 7 | |
21 | 8 | eleq2i 2693 | . . . . . . . . . . 11 |
22 | 21 | biimpi 206 | . . . . . . . . . 10 |
23 | velsn 4193 | . . . . . . . . . 10 | |
24 | 22, 23 | sylib 208 | . . . . . . . . 9 |
25 | 24 | fveq2d 6195 | . . . . . . . 8 |
26 | 25 | adantl 482 | . . . . . . 7 |
27 | 24 | fveq2d 6195 | . . . . . . . 8 |
28 | 27 | adantl 482 | . . . . . . 7 |
29 | 20, 26, 28 | 3eqtr4d 2666 | . . . . . 6 |
30 | 29 | adantll 750 | . . . . 5 |
31 | 30 | ralrimiva 2966 | . . . 4 |
32 | 31 | ex 450 | . . 3 |
33 | 19, 32 | impbid 202 | . 2 |
34 | 4, 33 | bitrd 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 csn 4177 wfn 5883 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-8 1992 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-pow 4843 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-csb 3534 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-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: fsneqrn 39403 unirnmapsn 39406 |
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