Theorem relpfr 45554

Description: If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7341 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 23	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))
2		relpf 45550	. . 3 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3		ffun 6709	. . 3 ⊢ (𝐻:𝐴⟶𝐵 → Fun 𝐻)
4		vex 3467	. . . 4 ⊢ 𝑥 ∈ V
5	4	funimaex 6624	. . 3 ⊢ (Fun 𝐻 → (𝐻 “ 𝑥) ∈ V)
6	2, 3, 5	3syl 19	. 2 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝐻 “ 𝑥) ∈ V)
7	1, 6	relpfrlem 45553	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463   Fr wfr 5612   “ cima 5665  Fun wfun 6531  ⟶wf 6533   RelPres wrelp 45542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-fr 5615  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-relp 45543

This theorem is referenced by:  wffr  45561