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Theorem fpmd 40284
Description: A total function is a partial function. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
fpmd.a (𝜑𝐴𝑉)
fpmd.b (𝜑𝐵𝑊)
fpmd.c (𝜑𝐶𝐴)
fpmd.f (𝜑𝐹:𝐶𝐵)
Assertion
Ref Expression
fpmd (𝜑𝐹 ∈ (𝐵pm 𝐴))

Proof of Theorem fpmd
StepHypRef Expression
1 fpmd.b . 2 (𝜑𝐵𝑊)
2 fpmd.a . 2 (𝜑𝐴𝑉)
3 fpmd.f . 2 (𝜑𝐹:𝐶𝐵)
4 fpmd.c . 2 (𝜑𝐶𝐴)
5 elpm2r 8141 . 2 (((𝐵𝑊𝐴𝑉) ∧ (𝐹:𝐶𝐵𝐶𝐴)) → 𝐹 ∈ (𝐵pm 𝐴))
61, 2, 3, 4, 5syl22anc 874 1 (𝜑𝐹 ∈ (𝐵pm 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  wss 3799  wf 6120  (class class class)co 6906  pm cpm 8124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-pm 8126
This theorem is referenced by:  xlimbr  40849  fuzxrpmcn  40850
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