Theorem relpfr 44987

Description: If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7271 for a bidirectional statement. A more general version of Lemma I.9.9 of [Kunen2] p. 47. (Contributed by Eric Schmidt, 11-Oct-2025.)

Assertion

Ref	Expression
relpfr	⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Proof of Theorem relpfr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))
2		relpf 44983	. . 3 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3		ffun 6649	. . 3 ⊢ (𝐻:𝐴⟶𝐵 → Fun 𝐻)
4		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
5	4	funimaex 6564	. . 3 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V)
6	2, 3, 5	3syl 18	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V)
7	1, 6	relpfrlem 44986	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436   Fr wfr 5561   “ cima 5614  Fun wfun 6470  ⟶wf 6472   RelPres wrelp 44975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-fr 5564  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-relp 44976

This theorem is referenced by:  wffr  44994