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Theorem fphpdo 36195
Description: Pigeonhole principle for sets of real numbers with implicit output reordering. (Contributed by Stefan O'Rear, 12-Sep-2014.)
Hypotheses
Ref Expression
fphpdo.1 (𝜑𝐴 ⊆ ℝ)
fphpdo.2 (𝜑𝐵 ∈ V)
fphpdo.3 (𝜑𝐵𝐴)
fphpdo.4 ((𝜑𝑧𝐴) → 𝐶𝐵)
fphpdo.5 (𝑧 = 𝑥𝐶 = 𝐷)
fphpdo.6 (𝑧 = 𝑦𝐶 = 𝐸)
Assertion
Ref Expression
fphpdo (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑧,𝐵   𝑥,𝐶,𝑦   𝑦,𝐷,𝑧   𝑥,𝐸,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑧)   𝐷(𝑥)   𝐸(𝑦)

Proof of Theorem fphpdo
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fphpdo.3 . . 3 (𝜑𝐵𝐴)
2 fphpdo.4 . . . . 5 ((𝜑𝑧𝐴) → 𝐶𝐵)
3 eqid 2609 . . . . 5 (𝑧𝐴𝐶) = (𝑧𝐴𝐶)
42, 3fmptd 6277 . . . 4 (𝜑 → (𝑧𝐴𝐶):𝐴𝐵)
54ffvelrnda 6252 . . 3 ((𝜑𝑏𝐴) → ((𝑧𝐴𝐶)‘𝑏) ∈ 𝐵)
6 fveq2 6088 . . 3 (𝑏 = 𝑐 → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
71, 5, 6fphpd 36194 . 2 (𝜑 → ∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
8 fphpdo.1 . . . . . . . . . 10 (𝜑𝐴 ⊆ ℝ)
98sselda 3567 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑏 ∈ ℝ)
109adantrr 748 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏 ∈ ℝ)
1110adantr 479 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑏 ∈ ℝ)
128sselda 3567 . . . . . . . . 9 ((𝜑𝑐𝐴) → 𝑐 ∈ ℝ)
1312adantrl 747 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐 ∈ ℝ)
1413adantr 479 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑐 ∈ ℝ)
1511, 14lttri2d 10027 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 ↔ (𝑏 < 𝑐𝑐 < 𝑏)))
16 simprl 789 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏𝐴)
1716ad2antrr 757 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏𝐴)
18 simprr 791 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐𝐴)
1918ad2antrr 757 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑐𝐴)
20 simpr 475 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏 < 𝑐)
21 simplr 787 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
22 breq1 4580 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (𝑥 < 𝑦𝑏 < 𝑦))
23 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑏))
2423eqeq1d 2611 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)))
2522, 24anbi12d 742 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦))))
26 breq2 4581 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝑏 < 𝑦𝑏 < 𝑐))
27 fveq2 6088 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑐))
2827eqeq2d 2619 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
2926, 28anbi12d 742 . . . . . . . . . . 11 (𝑦 = 𝑐 → ((𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))))
3025, 29rspc2ev 3294 . . . . . . . . . 10 ((𝑏𝐴𝑐𝐴 ∧ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3117, 19, 20, 21, 30syl112anc 1321 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3231ex 448 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏 < 𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
3318ad2antrr 757 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐𝐴)
3416ad2antrr 757 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑏𝐴)
35 simpr 475 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐 < 𝑏)
36 simplr 787 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
3736eqcomd 2615 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))
38 breq1 4580 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (𝑥 < 𝑦𝑐 < 𝑦))
39 fveq2 6088 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑐))
4039eqeq1d 2611 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)))
4138, 40anbi12d 742 . . . . . . . . . . 11 (𝑥 = 𝑐 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦))))
42 breq2 4581 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑐 < 𝑦𝑐 < 𝑏))
43 fveq2 6088 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑏))
4443eqeq2d 2619 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏)))
4542, 44anbi12d 742 . . . . . . . . . . 11 (𝑦 = 𝑏 → ((𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))))
4641, 45rspc2ev 3294 . . . . . . . . . 10 ((𝑐𝐴𝑏𝐴 ∧ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4733, 34, 35, 37, 46syl112anc 1321 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4847ex 448 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑐 < 𝑏 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
4932, 48jaod 393 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
50 simplr 787 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
51 eleq1 2675 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
5251anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝜑𝑧𝐴) ↔ (𝜑𝑥𝐴)))
53 fphpdo.5 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥𝐶 = 𝐷)
5453eleq1d 2671 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝐶𝐵𝐷𝐵))
5552, 54imbi12d 332 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑥𝐴) → 𝐷𝐵)))
5655, 2chvarv 2250 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐷𝐵)
5756adantr 479 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐷𝐵)
5853, 3fvmptg 6174 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐷𝐵) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
5950, 57, 58syl2anc 690 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
60 simpr 475 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
61 eleq1 2675 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6261anbi2d 735 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝜑𝑧𝐴) ↔ (𝜑𝑦𝐴)))
63 fphpdo.6 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦𝐶 = 𝐸)
6463eleq1d 2671 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝐶𝐵𝐸𝐵))
6562, 64imbi12d 332 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑦𝐴) → 𝐸𝐵)))
6665, 2chvarv 2250 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → 𝐸𝐵)
6766adantlr 746 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐸𝐵)
6863, 3fvmptg 6174 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐸𝐵) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
6960, 67, 68syl2anc 690 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
7059, 69eqeq12d 2624 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ 𝐷 = 𝐸))
7170biimpd 217 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) → 𝐷 = 𝐸))
7271anim2d 586 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → (𝑥 < 𝑦𝐷 = 𝐸)))
7372reximdva 2999 . . . . . . . . 9 ((𝜑𝑥𝐴) → (∃𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7473reximdva 2999 . . . . . . . 8 (𝜑 → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7574ad2antrr 757 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7649, 75syld 45 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7715, 76sylbid 228 . . . . 5 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7877expimpd 626 . . . 4 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐) ∧ 𝑏𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7978ancomsd 468 . . 3 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
8079rexlimdvva 3019 . 2 (𝜑 → (∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
817, 80mpd 15 1 (𝜑 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  Vcvv 3172  wss 3539   class class class wbr 4577  cmpt 4637  cfv 5790  csdm 7817  cr 9791   < clt 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-resscn 9849  ax-pre-lttri 9866  ax-pre-lttrn 9867
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-po 4949  df-so 4950  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-ltxr 9935
This theorem is referenced by:  irrapxlem1  36200
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