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Theorem metf1o 32611
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 5934 . . . . . . 7 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
2 ffvelrn 6149 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
32ex 448 . . . . . . . 8 (𝐹:𝑌𝑋 → (𝑥𝑌 → (𝐹𝑥) ∈ 𝑋))
4 ffvelrn 6149 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑦𝑌) → (𝐹𝑦) ∈ 𝑋)
54ex 448 . . . . . . . 8 (𝐹:𝑌𝑋 → (𝑦𝑌 → (𝐹𝑦) ∈ 𝑋))
63, 5anim12d 583 . . . . . . 7 (𝐹:𝑌𝑋 → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋)))
71, 6syl 17 . . . . . 6 (𝐹:𝑌1-1-onto𝑋 → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋)))
8 metcl 21849 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ)
983expib 1259 . . . . . 6 (𝑀 ∈ (Met‘𝑋) → (((𝐹𝑥) ∈ 𝑋 ∧ (𝐹𝑦) ∈ 𝑋) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
107, 9sylan9r 687 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
11103adant1 1071 . . . 4 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑥𝑌𝑦𝑌) → ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ))
1211ralrimivv 2857 . . 3 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ∀𝑥𝑌𝑦𝑌 ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ)
13 metf1o.2 . . . 4 𝑁 = (𝑥𝑌, 𝑦𝑌 ↦ ((𝐹𝑥)𝑀(𝐹𝑦)))
1413fmpt2 7001 . . 3 (∀𝑥𝑌𝑦𝑌 ((𝐹𝑥)𝑀(𝐹𝑦)) ∈ ℝ ↔ 𝑁:(𝑌 × 𝑌)⟶ℝ)
1512, 14sylib 206 . 2 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁:(𝑌 × 𝑌)⟶ℝ)
16 fveq2 5987 . . . . . . . . . . 11 (𝑥 = 𝑢 → (𝐹𝑥) = (𝐹𝑢))
1716oveq1d 6441 . . . . . . . . . 10 (𝑥 = 𝑢 → ((𝐹𝑥)𝑀(𝐹𝑦)) = ((𝐹𝑢)𝑀(𝐹𝑦)))
18 fveq2 5987 . . . . . . . . . . 11 (𝑦 = 𝑣 → (𝐹𝑦) = (𝐹𝑣))
1918oveq2d 6442 . . . . . . . . . 10 (𝑦 = 𝑣 → ((𝐹𝑢)𝑀(𝐹𝑦)) = ((𝐹𝑢)𝑀(𝐹𝑣)))
20 ovex 6454 . . . . . . . . . 10 ((𝐹𝑢)𝑀(𝐹𝑣)) ∈ V
2117, 19, 13, 20ovmpt2 6571 . . . . . . . . 9 ((𝑢𝑌𝑣𝑌) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
2221eqeq1d 2516 . . . . . . . 8 ((𝑢𝑌𝑣𝑌) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = 0))
2322adantl 480 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) = 0))
24 ffvelrn 6149 . . . . . . . . . . . . 13 ((𝐹:𝑌𝑋𝑢𝑌) → (𝐹𝑢) ∈ 𝑋)
2524ex 448 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝑢𝑌 → (𝐹𝑢) ∈ 𝑋))
26 ffvelrn 6149 . . . . . . . . . . . . 13 ((𝐹:𝑌𝑋𝑣𝑌) → (𝐹𝑣) ∈ 𝑋)
2726ex 448 . . . . . . . . . . . 12 (𝐹:𝑌𝑋 → (𝑣𝑌 → (𝐹𝑣) ∈ 𝑋))
2825, 27anim12d 583 . . . . . . . . . . 11 (𝐹:𝑌𝑋 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
291, 28syl 17 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋 → ((𝑢𝑌𝑣𝑌) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)))
3029imp 443 . . . . . . . . 9 ((𝐹:𝑌1-1-onto𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋))
3130adantll 745 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋))
32 meteq0 21856 . . . . . . . . . 10 ((𝑀 ∈ (Met‘𝑋) ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
33323expb 1257 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
3433adantlr 746 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
3531, 34syldan 485 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝐹𝑢)𝑀(𝐹𝑣)) = 0 ↔ (𝐹𝑢) = (𝐹𝑣)))
36 f1of1 5933 . . . . . . . . 9 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌1-1𝑋)
37 f1fveq 6297 . . . . . . . . 9 ((𝐹:𝑌1-1𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3836, 37sylan 486 . . . . . . . 8 ((𝐹:𝑌1-1-onto𝑋 ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
3938adantll 745 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝐹𝑢) = (𝐹𝑣) ↔ 𝑢 = 𝑣))
4023, 35, 393bitrd 292 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣))
41 ffvelrn 6149 . . . . . . . . . . . . . . 15 ((𝐹:𝑌𝑋𝑤𝑌) → (𝐹𝑤) ∈ 𝑋)
4241ex 448 . . . . . . . . . . . . . 14 (𝐹:𝑌𝑋 → (𝑤𝑌 → (𝐹𝑤) ∈ 𝑋))
4328, 42anim12d 583 . . . . . . . . . . . . 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 443 . . . . . . . . . . 11 ((𝐹:𝑌1-1-onto𝑋 ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋))
4645adantll 745 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋))
47 mettri2 21858 . . . . . . . . . . . . . . 15 ((𝑀 ∈ (Met‘𝑋) ∧ ((𝐹𝑤) ∈ 𝑋 ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
4847expcom 449 . . . . . . . . . . . . . 14 (((𝐹𝑤) ∈ 𝑋 ∧ (𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
49483expb 1257 . . . . . . . . . . . . 13 (((𝐹𝑤) ∈ 𝑋 ∧ ((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋)) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
5049ancoms 467 . . . . . . . . . . . 12 ((((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋) → (𝑀 ∈ (Met‘𝑋) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
5150impcom 444 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ (((𝐹𝑢) ∈ 𝑋 ∧ (𝐹𝑣) ∈ 𝑋) ∧ (𝐹𝑤) ∈ 𝑋)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5251adantlr 746 . . . . . . . . . 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 485 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ ((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌)) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5453anassrs 677 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
5521adantr 479 . . . . . . . . . 10 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑢𝑁𝑣) = ((𝐹𝑢)𝑀(𝐹𝑣)))
56 fveq2 5987 . . . . . . . . . . . . . . 15 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
5756oveq1d 6441 . . . . . . . . . . . . . 14 (𝑥 = 𝑤 → ((𝐹𝑥)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑦)))
58 fveq2 5987 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (𝐹𝑦) = (𝐹𝑢))
5958oveq2d 6442 . . . . . . . . . . . . . 14 (𝑦 = 𝑢 → ((𝐹𝑤)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑢)))
60 ovex 6454 . . . . . . . . . . . . . 14 ((𝐹𝑤)𝑀(𝐹𝑢)) ∈ V
6157, 59, 13, 60ovmpt2 6571 . . . . . . . . . . . . 13 ((𝑤𝑌𝑢𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6261ancoms 467 . . . . . . . . . . . 12 ((𝑢𝑌𝑤𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6362adantlr 746 . . . . . . . . . . 11 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑤𝑁𝑢) = ((𝐹𝑤)𝑀(𝐹𝑢)))
6418oveq2d 6442 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 → ((𝐹𝑤)𝑀(𝐹𝑦)) = ((𝐹𝑤)𝑀(𝐹𝑣)))
65 ovex 6454 . . . . . . . . . . . . . 14 ((𝐹𝑤)𝑀(𝐹𝑣)) ∈ V
6657, 64, 13, 65ovmpt2 6571 . . . . . . . . . . . . 13 ((𝑤𝑌𝑣𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6766ancoms 467 . . . . . . . . . . . 12 ((𝑣𝑌𝑤𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6867adantll 745 . . . . . . . . . . 11 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → (𝑤𝑁𝑣) = ((𝐹𝑤)𝑀(𝐹𝑣)))
6963, 68oveq12d 6444 . . . . . . . . . 10 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) = (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣))))
7055, 69breq12d 4494 . . . . . . . . 9 (((𝑢𝑌𝑣𝑌) ∧ 𝑤𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
7170adantll 745 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → ((𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)) ↔ ((𝐹𝑢)𝑀(𝐹𝑣)) ≤ (((𝐹𝑤)𝑀(𝐹𝑢)) + ((𝐹𝑤)𝑀(𝐹𝑣)))))
7254, 71mpbird 245 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) ∧ 𝑤𝑌) → (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
7372ralrimiva 2853 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))
7440, 73jca 552 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
75743adantl1 1209 . . . 4 (((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) ∧ (𝑢𝑌𝑣𝑌)) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
7675ex 448 . . 3 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ((𝑢𝑌𝑣𝑌) → (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣)))))
7776ralrimivv 2857 . 2 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → ∀𝑢𝑌𝑣𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))
78 ismet 21840 . . 3 (𝑌𝐴 → (𝑁 ∈ (Met‘𝑌) ↔ (𝑁:(𝑌 × 𝑌)⟶ℝ ∧ ∀𝑢𝑌𝑣𝑌 (((𝑢𝑁𝑣) = 0 ↔ 𝑢 = 𝑣) ∧ ∀𝑤𝑌 (𝑢𝑁𝑣) ≤ ((𝑤𝑁𝑢) + (𝑤𝑁𝑣))))))
79783ad2ant1 1074 . 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 958 1 ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁 ∈ (Met‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800   class class class wbr 4481   × cxp 4930  wf 5685  1-1wf1 5686  1-1-ontowf1o 5688  cfv 5689  (class class class)co 6426  cmpt2 6428  cr 9690  0cc0 9691   + caddc 9694  cle 9830  Metcme 19457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-mulcl 9753  ax-i2m1 9759
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-1st 6934  df-2nd 6935  df-er 7505  df-map 7622  df-en 7718  df-dom 7719  df-sdom 7720  df-pnf 9831  df-mnf 9832  df-xr 9833  df-xadd 11689  df-xmet 19464  df-met 19465
This theorem is referenced by: (None)
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