Theorem relpfr 45663

Description: If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7340 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 45659	. . 3 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3		ffun 6708	. . 3 ⊢ (𝐻:𝐴⟶𝐵 → Fun 𝐻)
4		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
5	4	funimaex 6623	. . 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 45662	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   Fr wfr 5611   “ cima 5664  Fun wfun 6530  ⟶wf 6532   RelPres wrelp 45651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-fr 5614  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-relp 45652

This theorem is referenced by:  wffr  45670