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Theorem metf1o 34898
Description: Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
metf1o.2 𝑁 = (𝑥𝑌, 𝑦𝑌 ↦ ((𝐹𝑥)𝑀(𝐹𝑦)))
Assertion
Ref Expression
metf1o ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁 ∈ (Met‘𝑌))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem metf1o
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of 6612 . . . . . . 7 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2 ffvelrn 6845 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
32ex 413 . . . . . . . 8 (𝐹:𝑌𝑋 → (𝑥𝑌 → (𝐹𝑥) ∈ 𝑋))
4 ffvelrn 6845 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑦𝑌) → (𝐹𝑦) ∈ 𝑋)
54ex 413 . . . . . . . 8 (𝐹:𝑌𝑋 → (𝑦𝑌 → (𝐹𝑦) ∈ 𝑋))
63, 5anim12d 608 . . . . . . 7 (𝐹:𝑌𝑋 → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋)))
71, 6syl 17 . . . . . 6 (𝐹:𝑌1-1-onto𝑋 → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋)))
8 metcl 22857 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ)
983expib 1116 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → (((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
107, 9sylan9r 509 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
11103adant1 1124 . . . 4 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
1211ralrimivv 3195 . . 3 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ∀𝑥𝑌𝑦𝑌 ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ)
13 metf1o.2 . . . 4 𝑁 = (𝑥𝑌, 𝑦𝑌 ↦ ((𝐹𝑥)𝑀(𝐹𝑦)))
1413fmpo 7757 . . 3 (∀𝑥𝑌𝑦𝑌 ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ ↔ 𝑁:(𝑌 × 𝑌)⟶ℝ)
1512, 14sylib 219 . 2 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁:(𝑌 × 𝑌)⟶ℝ)
16 fveq2 6667 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
1716oveq1d 7163 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)𝑀(𝐹𝑦)) = ((𝐹𝑢)𝑀(𝐹𝑦)))
18 fveq2 6667 . . . . . . . . . . 11 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
1918oveq2d 7164 . . . . . . . . . 10 (𝑦 = 𝑣 → ((𝐹𝑢)𝑀(𝐹𝑦)) = ((𝐹𝑢)𝑀(𝐹𝑣)))
20 ovex 7181 . . . . . . . . . 10 ((𝐹𝑢)𝑀(𝐹𝑣)) ∈ V
2117, 19, 13, 20ovmpo 7300 . . . . . . . . 9 ((𝑢𝑌𝑣𝑌) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
2221eqeq1d 2828 . . . . . . . 8 ((𝑢𝑌𝑣𝑌) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = 0))
2322adantl 482 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = 0))
24 ffvelrn 6845 . . . . . . . . . . . . 13 ((𝐹:𝑌𝑋𝑢𝑌) → (𝐹𝑢) ∈ 𝑋)
2524ex 413 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝑢𝑌 → (𝐹𝑢) ∈ 𝑋))
26 ffvelrn 6845 . . . . . . . . . . . . 13 ((𝐹:𝑌𝑋𝑣𝑌) → (𝐹𝑣) ∈ 𝑋)
2726ex 413 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝑣𝑌 → (𝐹𝑣) ∈ 𝑋))
2825, 27anim12d 608 . . . . . . . . . . 11 (𝐹:𝑌𝑋 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
291, 28syl 17 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
3029imp 407 . . . . . . . . 9 ((𝐹:𝑌1-1-onto𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋))
3130adantll 710 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋))
32 meteq0 22864 . . . . . . . . . 10 ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
33323expb 1114 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
3433adantlr 711 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
3531, 34syldan 591 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
36 f1of1 6611 . . . . . . . . 9 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌1-1𝑋)
37 f1fveq 7014 . . . . . . . . 9 ((𝐹:𝑌1-1𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3836, 37sylan 580 . . . . . . . 8 ((𝐹:𝑌1-1-onto𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3938adantll 710 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
4023, 35, 393bitrd 306 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣))
41 ffvelrn 6845 . . . . . . . . . . . . . . 15 ((𝐹:𝑌𝑋𝑤𝑌) → (𝐹𝑤) ∈ 𝑋)
4241ex 413 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → (𝑤𝑌 → (𝐹𝑤) ∈ 𝑋))
4328, 42anim12d 608 . . . . . . . . . . . . 13 (𝐹:𝑌𝑋 → (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋)))
441, 43syl 17 . . . . . . . . . . . 12 (𝐹:𝑌1-1-onto𝑋 → (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋)))
4544imp 407 . . . . . . . . . . 11 ((𝐹:𝑌1-1-onto𝑋 ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋))
4645adantll 710 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋))
47 mettri2 22866 . . . . . . . . . . . . . . 15 ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹𝑤) ∈ 𝑋 ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
4847expcom 414 . . . . . . . . . . . . . 14 (((𝐹𝑤) ∈ 𝑋 ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
49483expb 1114 . . . . . . . . . . . . 13 (((𝐹𝑤) ∈ 𝑋 ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
5049ancoms 459 . . . . . . . . . . . 12 ((((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
5150impcom 408 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5251adantlr 711 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5346, 52syldan 591 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5453anassrs 468 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5521adantr 481 . . . . . . . . . 10 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
56 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
5756oveq1d 7163 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → ((𝐹𝑥)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑦)))
58 fveq2 6667 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝐹𝑦) = (𝐹𝑢))
5958oveq2d 7164 . . . . . . . . . . . . . 14 (𝑦 = 𝑢 → ((𝐹𝑤)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑢)))
60 ovex 7181 . . . . . . . . . . . . . 14 ((𝐹𝑤)𝑀(𝐹𝑢)) ∈ V
6157, 59, 13, 60ovmpo 7300 . . . . . . . . . . . . 13 ((𝑤𝑌𝑢𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6261ancoms 459 . . . . . . . . . . . 12 ((𝑢𝑌𝑤𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6362adantlr 711 . . . . . . . . . . 11 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6418oveq2d 7164 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝐹𝑤)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑣)))
65 ovex 7181 . . . . . . . . . . . . . 14 ((𝐹𝑤)𝑀(𝐹𝑣)) ∈ V
6657, 64, 13, 65ovmpo 7300 . . . . . . . . . . . . 13 ((𝑤𝑌𝑣𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6766ancoms 459 . . . . . . . . . . . 12 ((𝑣𝑌𝑤𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6867adantll 710 . . . . . . . . . . 11 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6963, 68oveq12d 7166 . . . . . . . . . 10 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) = (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
7055, 69breq12d 5076 . . . . . . . . 9 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
7170adantll 710 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
7254, 71mpbird 258 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
7372ralrimiva 3187 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
7440, 73jca 512 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
75743adantl1 1160 . . . 4 (((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
7675ex 413 . . 3 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑢𝑌𝑣𝑌) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))))
7776ralrimivv 3195 . 2 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ∀𝑢𝑌𝑣𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
78 ismet 22848 . . 3 (𝑌𝐴 → (𝑁 ∈ (Met‘𝑌) ↔ (𝑁:(𝑌 × 𝑌)⟶ℝ ∧ ∀𝑢𝑌𝑣𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))))
79783ad2ant1 1127 . 2 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → (𝑁 ∈ (Met‘𝑌) ↔ (𝑁:(𝑌 × 𝑌)⟶ℝ ∧ ∀𝑢𝑌𝑣𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))))
8015, 77, 79mpbir2and 709 1 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁 ∈ (Met‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143   class class class wbr 5063   × cxp 5552  wf 6348  1-1wf1 6349  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  cmpo 7150  cr 10525  0cc0 10526   + caddc 10529  cle 10665  Metcmet 20447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-mulcl 10588  ax-i2m1 10594
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-xadd 12498  df-xmet 20454  df-met 20455
This theorem is referenced by: (None)
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