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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffunALTV2 | Structured version Visualization version GIF version |
Description: Alternate definition of the function relation predicate, cf. dfdisjALTV2 37453. (Contributed by Peter Mazsa, 8-Feb-2018.) |
Ref | Expression |
---|---|
dffunALTV2 | ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-funALTV 37421 | . 2 ⊢ ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)) | |
2 | cnvrefrelcoss2 37276 | . . 3 ⊢ ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I ) | |
3 | 2 | anbi1i 624 | . 2 ⊢ (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹)) |
4 | 1, 3 | bitri 274 | 1 ⊢ ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ⊆ wss 3945 I cid 5567 Rel wrel 5675 ≀ ccoss 36912 CnvRefRel wcnvrefrel 36921 FunALTV wfunALTV 36943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5143 df-opab 5205 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-coss 37150 df-cnvrefrel 37266 df-funALTV 37421 |
This theorem is referenced by: dffunALTV3 37428 dffunALTV4 37429 dffunALTV5 37430 funALTVss 37438 |
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