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Theorem funALTVss 38025
Description: Subclass theorem for function. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) (Revised by Peter Mazsa, 22-Sep-2021.)
Assertion
Ref Expression
funALTVss (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))

Proof of Theorem funALTVss
StepHypRef Expression
1 cossss 37751 . . . 4 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
2 sstr2 3981 . . . 4 ( ≀ 𝐴 ⊆ ≀ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
31, 2syl 17 . . 3 (𝐴𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
4 relss 5771 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 608 . 2 (𝐴𝐵 → (( ≀ 𝐵 ⊆ I ∧ Rel 𝐵) → ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴)))
6 dffunALTV2 38014 . 2 ( FunALTV 𝐵 ↔ ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵))
7 dffunALTV2 38014 . 2 ( FunALTV 𝐴 ↔ ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴))
85, 6, 73imtr4g 296 1 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3940   I cid 5563  Rel wrel 5671  ccoss 37499   FunALTV wfunALTV 37530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-coss 37737  df-cnvrefrel 37853  df-funALTV 38008
This theorem is referenced by:  funALTVeq  38026  disjss  38057
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