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Theorem fphpdo 37698
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 2651 . . . . 5 (𝑧𝐴𝐶) = (𝑧𝐴𝐶)
42, 3fmptd 6425 . . . 4 (𝜑 → (𝑧𝐴𝐶):𝐴𝐵)
54ffvelrnda 6399 . . 3 ((𝜑𝑏𝐴) → ((𝑧𝐴𝐶)‘𝑏) ∈ 𝐵)
6 fveq2 6229 . . 3 (𝑏 = 𝑐 → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
71, 5, 6fphpd 37697 . 2 (𝜑 → ∃𝑏𝐴𝑐𝐴 (𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
8 fphpdo.1 . . . . . . . . . 10 (𝜑𝐴 ⊆ ℝ)
98sselda 3636 . . . . . . . . 9 ((𝜑𝑏𝐴) → 𝑏 ∈ ℝ)
109adantrr 753 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏 ∈ ℝ)
1110adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑏 ∈ ℝ)
128sselda 3636 . . . . . . . . 9 ((𝜑𝑐𝐴) → 𝑐 ∈ ℝ)
1312adantrl 752 . . . . . . . 8 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐 ∈ ℝ)
1413adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → 𝑐 ∈ ℝ)
1511, 14lttri2d 10214 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 ↔ (𝑏 < 𝑐𝑐 < 𝑏)))
16 simprl 809 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑏𝐴)
1716ad2antrr 762 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏𝐴)
18 simprr 811 . . . . . . . . . . 11 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → 𝑐𝐴)
1918ad2antrr 762 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑐𝐴)
20 simpr 476 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → 𝑏 < 𝑐)
21 simplr 807 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
22 breq1 4688 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (𝑥 < 𝑦𝑏 < 𝑦))
23 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝑏 → ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑏))
2423eqeq1d 2653 . . . . . . . . . . . 12 (𝑥 = 𝑏 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)))
2522, 24anbi12d 747 . . . . . . . . . . 11 (𝑥 = 𝑏 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦))))
26 breq2 4689 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (𝑏 < 𝑦𝑏 < 𝑐))
27 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝑐 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑐))
2827eqeq2d 2661 . . . . . . . . . . . 12 (𝑦 = 𝑐 → (((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)))
2926, 28anbi12d 747 . . . . . . . . . . 11 (𝑦 = 𝑐 → ((𝑏 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))))
3025, 29rspc2ev 3355 . . . . . . . . . 10 ((𝑏𝐴𝑐𝐴 ∧ (𝑏 < 𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3117, 19, 20, 21, 30syl112anc 1370 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑏 < 𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
3231ex 449 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏 < 𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
3318ad2antrr 762 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐𝐴)
3416ad2antrr 762 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑏𝐴)
35 simpr 476 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → 𝑐 < 𝑏)
36 simplr 807 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐))
3736eqcomd 2657 . . . . . . . . . 10 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))
38 breq1 4688 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (𝑥 < 𝑦𝑐 < 𝑦))
39 fveq2 6229 . . . . . . . . . . . . 13 (𝑥 = 𝑐 → ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑐))
4039eqeq1d 2653 . . . . . . . . . . . 12 (𝑥 = 𝑐 → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)))
4138, 40anbi12d 747 . . . . . . . . . . 11 (𝑥 = 𝑐 → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦))))
42 breq2 4689 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (𝑐 < 𝑦𝑐 < 𝑏))
43 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝑏 → ((𝑧𝐴𝐶)‘𝑦) = ((𝑧𝐴𝐶)‘𝑏))
4443eqeq2d 2661 . . . . . . . . . . . 12 (𝑦 = 𝑏 → (((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦) ↔ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏)))
4542, 44anbi12d 747 . . . . . . . . . . 11 (𝑦 = 𝑏 → ((𝑐 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑦)) ↔ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))))
4641, 45rspc2ev 3355 . . . . . . . . . 10 ((𝑐𝐴𝑏𝐴 ∧ (𝑐 < 𝑏 ∧ ((𝑧𝐴𝐶)‘𝑐) = ((𝑧𝐴𝐶)‘𝑏))) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4733, 34, 35, 37, 46syl112anc 1370 . . . . . . . . 9 ((((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) ∧ 𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)))
4847ex 449 . . . . . . . 8 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑐 < 𝑏 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
4932, 48jaod 394 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦))))
50 simplr 807 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑥𝐴)
51 eleq1 2718 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
5251anbi2d 740 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → ((𝜑𝑧𝐴) ↔ (𝜑𝑥𝐴)))
53 fphpdo.5 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑥𝐶 = 𝐷)
5453eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑥 → (𝐶𝐵𝐷𝐵))
5552, 54imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑥𝐴) → 𝐷𝐵)))
5655, 2chvarv 2299 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐷𝐵)
5756adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐷𝐵)
5853, 3fvmptg 6319 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐷𝐵) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
5950, 57, 58syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑥) = 𝐷)
60 simpr 476 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝑦𝐴)
61 eleq1 2718 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
6261anbi2d 740 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → ((𝜑𝑧𝐴) ↔ (𝜑𝑦𝐴)))
63 fphpdo.6 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦𝐶 = 𝐸)
6463eleq1d 2715 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑦 → (𝐶𝐵𝐸𝐵))
6562, 64imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (((𝜑𝑧𝐴) → 𝐶𝐵) ↔ ((𝜑𝑦𝐴) → 𝐸𝐵)))
6665, 2chvarv 2299 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → 𝐸𝐵)
6766adantlr 751 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → 𝐸𝐵)
6863, 3fvmptg 6319 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐸𝐵) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
6960, 67, 68syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑧𝐴𝐶)‘𝑦) = 𝐸)
7059, 69eqeq12d 2666 . . . . . . . . . . . 12 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) ↔ 𝐷 = 𝐸))
7170biimpd 219 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → (((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦) → 𝐷 = 𝐸))
7271anim2d 588 . . . . . . . . . 10 (((𝜑𝑥𝐴) ∧ 𝑦𝐴) → ((𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → (𝑥 < 𝑦𝐷 = 𝐸)))
7372reximdva 3046 . . . . . . . . 9 ((𝜑𝑥𝐴) → (∃𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7473reximdva 3046 . . . . . . . 8 (𝜑 → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7574ad2antrr 762 . . . . . . 7 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦 ∧ ((𝑧𝐴𝐶)‘𝑥) = ((𝑧𝐴𝐶)‘𝑦)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7649, 75syld 47 . . . . . 6 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ((𝑏 < 𝑐𝑐 < 𝑏) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7715, 76sylbid 230 . . . . 5 (((𝜑 ∧ (𝑏𝐴𝑐𝐴)) ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → (𝑏𝑐 → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7877expimpd 628 . . . 4 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐) ∧ 𝑏𝑐) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
7978ancomsd 469 . . 3 ((𝜑 ∧ (𝑏𝐴𝑐𝐴)) → ((𝑏𝑐 ∧ ((𝑧𝐴𝐶)‘𝑏) = ((𝑧𝐴𝐶)‘𝑐)) → ∃𝑥𝐴𝑦𝐴 (𝑥 < 𝑦𝐷 = 𝐸)))
8079rexlimdvva 3067 . 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 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  cfv 5926  csdm 7996  cr 9973   < clt 10112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117
This theorem is referenced by:  irrapxlem1  37703
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