Theorem relpfr 44964

Description: If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7369 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 44960	. . 3 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → 𝐻:𝐴⟶𝐵)
3		ffun 6747	. . 3 ⊢ (𝐻:𝐴⟶𝐵 → Fun 𝐻)
4		vex 3485	. . . 4 ⊢ 𝑥 ∈ V
5	4	funimaex 6663	. . 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 44963	1 ⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3481   Fr wfr 5642   “ cima 5696  Fun wfun 6563  ⟶wf 6565   RelPres wrelp 44952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-rep 5288  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-id 5587  df-fr 5645  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-fv 6577  df-relp 44953

This theorem is referenced by:  wffr  44967