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Theorem funALTVss 37211
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 36937 . . . 4 (𝐴𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
2 sstr2 3955 . . . 4 ( ≀ 𝐴 ⊆ ≀ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
31, 2syl 17 . . 3 (𝐴𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
4 relss 5741 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 610 . 2 (𝐴𝐵 → (( ≀ 𝐵 ⊆ I ∧ Rel 𝐵) → ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴)))
6 dffunALTV2 37200 . 2 ( FunALTV 𝐵 ↔ ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵))
7 dffunALTV2 37200 . 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 397  wss 3914   I cid 5534  Rel wrel 5642  ccoss 36684   FunALTV wfunALTV 36715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-coss 36923  df-cnvrefrel 37039  df-funALTV 37194
This theorem is referenced by:  funALTVeq  37212  disjss  37243
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