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Theorem rnghmval 44904
 Description: The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
Hypotheses
Ref Expression
isrnghm.b 𝐵 = (Base‘𝑅)
isrnghm.t · = (.r𝑅)
isrnghm.m = (.r𝑆)
rnghmval.c 𝐶 = (Base‘𝑆)
rnghmval.p + = (+g𝑅)
rnghmval.a = (+g𝑆)
Assertion
Ref Expression
rnghmval ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
Distinct variable groups:   𝐵,𝑓,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝐶,𝑓
Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦,𝑓)   (𝑥,𝑦,𝑓)   · (𝑥,𝑦,𝑓)   (𝑥,𝑦,𝑓)

Proof of Theorem rnghmval
Dummy variables 𝑟 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rnghomo 44900 . . 3 RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
21a1i 11 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))}))
3 fveq2 6658 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 isrnghm.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4eqtr4di 2811 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65csbeq1d 3809 . . . . 5 (𝑟 = 𝑅(Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
7 fveq2 6658 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
8 rnghmval.c . . . . . . . 8 𝐶 = (Base‘𝑆)
97, 8eqtr4di 2811 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐶)
109csbeq1d 3809 . . . . . 6 (𝑠 = 𝑆(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
1110csbeq2dv 3812 . . . . 5 (𝑠 = 𝑆𝐵 / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
126, 11sylan9eq 2813 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
1312adantl 485 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
144fvexi 6672 . . . . . 6 𝐵 ∈ V
158fvexi 6672 . . . . . 6 𝐶 ∈ V
16 oveq12 7159 . . . . . . . 8 ((𝑤 = 𝐶𝑣 = 𝐵) → (𝑤m 𝑣) = (𝐶m 𝐵))
1716ancoms 462 . . . . . . 7 ((𝑣 = 𝐵𝑤 = 𝐶) → (𝑤m 𝑣) = (𝐶m 𝐵))
18 raleq 3323 . . . . . . . . 9 (𝑣 = 𝐵 → (∀𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
1918raleqbi1dv 3321 . . . . . . . 8 (𝑣 = 𝐵 → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
2019adantr 484 . . . . . . 7 ((𝑣 = 𝐵𝑤 = 𝐶) → (∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))))
2117, 20rabeqbidv 3398 . . . . . 6 ((𝑣 = 𝐵𝑤 = 𝐶) → {𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})
2214, 15, 21csbie2 3844 . . . . 5 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))}
23 fveq2 6658 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (+g𝑟) = (+g𝑅))
24 rnghmval.p . . . . . . . . . . . 12 + = (+g𝑅)
2523, 24eqtr4di 2811 . . . . . . . . . . 11 (𝑟 = 𝑅 → (+g𝑟) = + )
2625oveqdr 7178 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(+g𝑟)𝑦) = (𝑥 + 𝑦))
2726fveq2d 6662 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(+g𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
28 fveq2 6658 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
29 rnghmval.a . . . . . . . . . . . 12 = (+g𝑆)
3028, 29eqtr4di 2811 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = )
3130adantl 485 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (+g𝑠) = )
3231oveqd 7167 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
3327, 32eqeq12d 2774 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
34 fveq2 6658 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
35 isrnghm.t . . . . . . . . . . . 12 · = (.r𝑅)
3634, 35eqtr4di 2811 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
3736oveqdr 7178 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
3837fveq2d 6662 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑓‘(𝑥(.r𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
39 fveq2 6658 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (.r𝑠) = (.r𝑆))
40 isrnghm.m . . . . . . . . . . . 12 = (.r𝑆)
4139, 40eqtr4di 2811 . . . . . . . . . . 11 (𝑠 = 𝑆 → (.r𝑠) = )
4241adantl 485 . . . . . . . . . 10 ((𝑟 = 𝑅𝑠 = 𝑆) → (.r𝑠) = )
4342oveqd 7167 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
4438, 43eqeq12d 2774 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
4533, 44anbi12d 633 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))))
46452ralbidv 3128 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))) ↔ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))))
4746rabbidv 3392 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
4822, 47syl5eq 2805 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
4948adantl 485 . . 3 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → 𝐵 / 𝑣𝐶 / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
5013, 49eqtrd 2793 . 2 (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝑟 = 𝑅𝑠 = 𝑆)) → (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤m 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))} = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
51 simpl 486 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑅 ∈ Rng)
52 simpr 488 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → 𝑆 ∈ Rng)
53 ovex 7183 . . . 4 (𝐶m 𝐵) ∈ V
5453rabex 5202 . . 3 {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ∈ V
5554a1i 11 . 2 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))} ∈ V)
562, 50, 51, 52, 55ovmpod 7297 1 ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074  Vcvv 3409  ⦋csb 3805  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416  Basecbs 16541  +gcplusg 16623  .rcmulr 16624  Rngcrng 44887   RngHomo crngh 44898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rnghomo 44900 This theorem is referenced by:  isrnghm  44905
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